Radu had an elder sister, who died in 2006 at age 14; he used to wear a t-shirt with a picture of her under the kit during the time he was a junior. He is the cousin of defender Andrei Radu, with whom he shares the same name.

Radu began playing football at age 8 with local Viitorul București, after his grandmother saw a newspaper announcement. He was transferred to FC Steaua București in 2008 for €12,000, and left the club for rival side Dinamo București three years later. At the beginning of 2013 he relocated to Italy, briefly joining Pergolettese before being signed by Internazionale in the mid-2013 and assigned to their junior squads.

On 14 May 2016, the last matchday of the Serie A season, Radu made his senior debut for Inter as a 72nd-minute substitute for Juan Pablo Carrizo in a 1–3 away defeat to Sassuolo. On 4 October, he penned a new four-year deal with the Nerazzurri which would run until June 2020.

Radu agreed to a one-year loan with second division side Avellino with the option to make the move permanent on 15 July 2017. He made his debut in the second round of the Coppa Italia, a 1–0 home win over Matera, on 6 August. One week later, Radu played in the third round of the same competition, a 3–1 away defeat to Hellas Verona. On 18 September, he made his Serie B debut for Avellino in a 1–1 home draw with Venezia. On 24 October, he kept his first clean sheet in a 1–0 home win over Pro Vercelli. Radu ended his loan with the Biancoverdi with 24 appearances, 5 clean sheets and 30 goals conceded.

On 29 June 2018, 21-year-old Radu was loaned to Genoa with an obligation to join the club permanently at the start of the following season. Genoa Chairman Enrico Preziosi claimed the subsequent transfer fee was €9 million, which made him the most expensive Romanian goalkeeper. However, according to the financial report of Internazionale, the fee was €8 million. The deal also reportedly included an undisclosed buy-back clause.

Radu was nominated for Gazeta Sporturilor’s 2018 Romanian Footballer of the Year award in late November. On 22 December, he saved one of the two penalty kicks awarded to Atalanta in a 3–1 home victory at the Stadio Luigi Ferraris. In January 2019, he was included by UEFA.com in a list of the fifty best young footballers to watch for that year. In spite of being coached by three different managers throughout the season, Radu maintained his place as first-choice goalkeeper and amassed 33 league appearances.

In June 2019, as stipulated Genoa agreed to buy Radu for €8 million, but he was immediately redeemed by Internazionale for a reported €12 million. The deal was officially announced in July 2019, and at the same time he returned to Genoa on another loan.

Radu represented Romania at under-21 level, featuring for the side in the 2019 UEFA European Championship qualifiers. He made his debut on 13 June 2017, in a 2–0 away win over Liechtenstein. Radu captained both of the group matches in September 2018, saving a penalty to preserve a win against Portugal and keeping a clean sheet in a victory over Bosnia and Herzegovina. The following month, Tricolorii mici went on to finish top of the group and thus qualified for the final tournament for the second time in their history.

Radu was called up for the first time to the Romania senior squad for the UEFA Euro 2020 qualifiers against Norway and Malta in June 2019. Later that month, he was a starter in all three group fixtures of the UEFA Under-21 Euro in Italy, conceding three goals and keeping a clean sheet in a draw against France which secured the first place for his nation. Romania were eventually eliminated by defending champions Germany in the semi-finals, after a 2–4 loss.

On 9 October 2020, Radu tested positive for COVID-19 amid its pandemic in Italy.