Bardi is a product of Livorno's youth system. He made his professional and Serie A debut on 16 May 2010, aged 18, starting the home game versus Parma, with the latter coming off 1–4 winners, when Livorno had already been relegated.
In January 2011 he was signed by Internazionale, on loan with an option to purchase half of the registration rights and he played for Inter's U-19 youth team until the end of the 2010–11 season. He replaced injured Raffaele Di Gennaro as first choice, winning 2011 Torneo di Viareggio. Inter excised the rights for €1.35 million in a 4-year contract but on 3 August 2011 returned Bardi on loan to Livorno for the 2011–12 season, in order to gain first team experience. On 27 August 2011 he made his Serie B debut playing as the first choice goalkeeper in the away match won 2–1 against Crotone. In the whole season Bardi was ahead original starter Alfonso De Lucia. In June 2012, Inter bought Bardi outright for another €4 million and on 14 July 2012, let him join Serie B team Novara Calcio on a loan deal. At Novara he was the first choice goalkeeper for the duration of the season and generated significant hype around himself with his good performances, making 35 appearances. After the expiration of the loan, Internazionale again loaned him out, this time back to newly promoted Serie A side Livorno Calcio.
Loan to ChievoVerona
On 27 June 2014, Bardi was loaned out to Serie A side A.C. Chievo Verona for the 2014–15 season. On 22 August he made his debut for Chievo Verona in the third round of Coppa Italia in a 1–0 away defeat against Pescara. On 30 August, Bardi made his Serie A debut in a 1–0 home defeat against Juventus. On 14 September he save a penalty from Gonzalo Higuain in a 1–0 away win over Napoli. On 26 October he save another penalty in a 2–1 home defeat against Genoa. Bardi play the first 9 Serie A match, but he became the second goalkeeper after the arrive of Albano Bizzarri. On 17 May he made the last appearances in a 2–0 away defeat against Torino. Bardi ended his loan to Chievo Verona with 11 appearances, kept 1 clean sheet and conceding 18 goals.
On 26 July 2015, he was signed by Espanyol with a season-long loan with option to buy, for replacing Kiko Casilla. On 3 December he made his debut for Espanyol in the round of 32 of the Copa del Rey in a 1–1 away draw against Levante. On 13 January, Bardi played in the round of 16 of the Copa del Rey in a 2–0 home defeat against Barcelona. At the end of January, Bardi was recall to Inter and he finish his loan to Espanyol with only 2 appearances.
Loan to Frosinone
On 28 January 2016, Bardi was signed with a 6-month loan from Serie A club Frosinone. He is the second goalkeeper after Nicola Leali. On 1 May, Bardi made his debut for Frosinone in a 3–3 away draw against Milan, in this match he save a penalty from Mario Balotelli. One week later he play in the last match of Serie A in 1–0 home defeat against Sassuolo. He followed the club relegated to Serie B, which Frosinone borrowed him for another season, with an option to sign outright. On 13 August he made played his first game of the season in the third round of Coppa Italia in a 2–0 away defeat against Pescara. On 27 August, Bardi played his first Serie B match for Frosinone in a 2–0 home win over Virtus Entella. The loan was renewed again on 13 July 2017.
