Keane worked his way up through the Manchester United academy, and made his first appearance for the under-18s on 21 April 2007 against Sheffield United, a game in which both sides had agreed to field their under-16 teams; at the age of 14, Keane came on as a substitute for goalscorer Tomos Roberts in the 77th minute of a 4–0 away win. His next appearance for the under-18s came almost two years later, when he started in central midfield in a 1–0 home defeat to Stoke City on 21 February 2009. Shortly before their 16th birthday earlier in the year, both Keanes were awarded three-year scholarships with Manchester United, which began in July 2009.

The following season, Keane became a regular starter for the under-18s, scoring 13 goals in 26 appearances in 2009–10, including all three goals in a 3–0 win at home to Manchester City; he also made several substitute appearances for the reserves. His reserve debut came as a substitute in a 1–0 league win away to Everton on 6 October 2009, and he scored his first goal five months later, in Manchester United’s penultimate group match in the Manchester Senior Cup against Stockport County on 8 March 2010; after coming on for Nicky Ajose on the hour mark, Keane scored 14 minutes later to give Manchester United a 1–0 win.

Despite being eligible for the Republic of Ireland through his father, Keane made his debut for the England under-16 team in a 3–1 win against Russia in 2009. He scored one goal in three appearances for the side. He was part of the England under-17 team that won the 2010 UEFA European Under-17 Football Championship, appearing in the final. He made a total of 15 appearances and scored three goals for the team. He then progressed to the under-19 side where he made six appearances and scored one goal.

Despite making just 17 appearances for the under-18s in 2010–11, Keane finished as the team’s top scorer with 16 goals, including eight in just six FA Youth Cup matches – three in the semi-final second leg against Chelsea and three more over the two legs of the final against Sheffield United. He also scored six goals in 10 appearances for the reserves; only a missed penalty denied him a hat-trick in a 5–1 win over Newcastle United on 16 December 2010, Ole Gunnar Solskjær’s final match as manager.

After being named as an unused substitute for a 2–1 defeat to Wolverhampton Wanderers in the Premier League on 5 February 2011 and a 3–0 win over Aldershot Town in the fourth round of the 2011–12 Football League Cup on 25 October 2011, Keane made his senior debut on 31 December 2011 in a 3–2 defeat at home to Blackburn Rovers in the Premier League, coming on as a late substitute for Rafael.

He received a call up to the under-21 side managed by Stuart Pearce in November 2011, and received praise from Manchester United reserve team manager Warren Joyce, with Joyce saying Keane deserved the call up after his good form for the reserve team. He made his first appearance during a 5–0 win against Iceland, coming off the bench to replace midfielder Josh McEachran in the 78th minute. He appeared in the following game four days later against Belgium, this time coming on as a substitute for Marvin Sordell. On 25 May 2012, he scored twice as England under-19s beat Slovenia 5–0 at Deepdale. In 2019, it was reported that Keane was set to declare for the Republic of Ireland after he met with Ireland manager Mick McCarthy. In February 2019, in Ipswich match notes, he confirmed that he wanted to play for Ireland and that he hoped to be included in Ireland squads in the coming months, confirming that he had declared for Ireland.

Keane was ever-present for the reserves in the league in 2011–12, scoring 14 times in 22 appearances, as well as four more in seven cup appearances. His best performance of the season came on 16 April 2012, when he scored four in a 6–3 win away to Newcastle United to confirm Manchester United as winners of the Premier Reserve League North. They again faced Aston Villa in the championship play-off, and again the match went to penalties, but this time Keane scored from his kick as United won 3–1. Keane was on the scoresheet again a week later as United made it a double with a 2–0 victory over Manchester City in the final of the Manchester Senior Cup.

A knee ligament injury suffered while on international duty with the England under-19s in May 2012 kept Keane out for the entire 2012–13 season. He made his return for the under-21s at home to Bolton Wanderers on 16 September 2013, scoring the first goal in a 4–1 win. Three more goals followed in the under-21s’ next two matches, against Bury and Bolton in the Manchester Senior Cup.

On 28 November 2013, Keane joined fellow Manchester United youngster Nick Powell on a one-month loan at Wigan Athletic. He made his debut at home to Derby County the following weekend, only to be replaced by Grant Holt at half-time with Wigan 3–0 down; they went on to lose the game 3–1. After failing to find the net in four appearances for Wigan, Keane returned to Manchester United in December 2013 having suffered a groin injury.

After recovering from injury, Keane made a loan move to Queens Park Rangers on 31 January 2014 until the end of the 2013–14 season. He made 10 appearances during the second half of the 2013–14 season as QPR won promotion to the Premier League through the Championship play-offs.

On 22 January 2015 Keane joined Sheffield Wednesday on loan for the remainder of the season. He made his Wednesday debut on 27 January in a 0–0 draw at Hillsborough against Birmingham City. His first senior goal came from the penalty spot to draw level in the Owls’ 1–1 home draw to Cardiff City on 7 February. He made 13 appearances during his loan at Sheffield Wednesday, scoring 3 goals.

On 8 July 2015, Keane signed on a season-long loan deal with Preston North End. On 31 December 2015, Manchester United recalled Keane from his loan. He made 22 appearances in all competitions during his loan spell at Deepdale, scoring twice.

On 30 August 2016, Keane signed a permanent deal with Hull City; the player moved to the Tigers on a three-year deal, reuniting with ex-Manchester United assistant manager Mike Phelan. He made his debut on 10 September 2016 away to Burnley when he came off the bench to replace Adama Diomande after 70 minutes. On 6 November 2016, during a match against Southampton, Keane suffered the second serious knee ligament injury of his career, putting him out of action until January 2018. He scored his first goal for the club on 10 April 2018, the final goal in a 0–5 away win against Burton Albion.

On 4 January 2019, Keane signed for Ipswich Town on loan until the end of the season. He made his debut for the club as a second-half substitute in a 0–1 FA Cup away loss to Accrington Stanley at the Crown Ground. He scored his first goal for the club on his home debut in a 1–0 win over Rotherham United at Portman Road, on 12 January 2019. He made 12 appearances for the Tractor Boys during his loan spell, scoring 3 goals.

On 20 August 2019, Keane signed with Ipswich Town permanently on a free transfer. He signed a one-year contract, with the option of an additional 12 months. He made his first appearance following his return to the club as a second-half substitute in a 0–0 draw with Doncaster Rovers at Portman Road on 14 September. He scored his first goal of the season on 8 October, netting the fourth goal in a 4–0 home win over Gillingham in a group stage match of the EFL Trophy. Keane scored his first league goal of the season on 7 December in a 1–1 draw with Coventry City.

With football suspended due to the COVID-19 pandemic and with his contract expiring, on 18 May 2020 the club announced they had decided against taking up the additional 12 month option in his contract or offer him a new deal – though the club left the door open for a potential new deal to be agreed once football had resumed.

On 9 October 2020, Keane signed a short-term contract with Wigan Athletic until 9 January 2021. He made his debut a day later, starting in a 0–3 loss to Crewe Alexandra. He scored his first goal for Wigan on 24 October, scoring the equalising goal in a 1–1 draw with Plymouth Argyle.