The post is largely ceremonial, but the vote is seen as an important moment for Mr Renzi, after the resignation of Giorgio Napolitano earlier this month.
Mr Napolitano, 89, stood down this month, citing “signs of fatigue”.
No result is expected before Saturday, as a winning candidate needs a two-thirds majority in initial voting.
More than 1,000 people were due to vote in the procedure, which began on Thursday, including both the Chamber of Deputies, the Senate and 58 regional representatives.
Mr Napolitano was applauded as he entered the chamber to cast his vote in the first round, having told journalists that Mr Mattarella was a person of “absolute loyalty and propriety”.
Mr Mattarella once served as defence minister and is a member of Mr Renzi’s centre-left Democratic Party (PD). His nomination was given unanimous support at a meeting of party voters, who number around 400.
However a candidate needs 673 votes to succeed in the first three rounds, and that is seen as unlikely. Italy’s RAI TV suggested Mr Mattarella could attract some 561 votes.
If voting extends to a fourth round on Saturday, a simple majority of 505 would be enough for him to become 12th president of the Italian republic.
Under the constitution, any Italian citizen over 50 can be nominated by MPs.
The role has the key power of appointing a prime minister, which in Italy is quite a frequent task, says BBC Rome correspondent James Reynolds. Mr Napolitano named five prime ministers during his eight-and-a-half years in office.
Mr Napolitano retired earlier this month, aged 89, saying poor health meant he was no longer able to do his job as he wished.
He had only agreed to serve a second term in an attempt to end political paralysis after inconclusive elections in 2013.
Mr Renzi’s party will need further backing from other parties, such as Nichi Vendola’s Left Ecology Freedom (SEL) Party, former members of Beppe Grillo’s anti-establishment Five Star Party, and centrist parties such Civic Choice.
However the centre-right Forza Italia Party of former Prime Minister Silvio Berlusconi is opposed to Mr Mattarella’s candidacy.
He was said to support another former premier, Giuliano Amato, and the ballot is secret, so rebel lawmakers could sabotage a vote, which happened when a large number of PD electors failed to support the party candidate two years ago.
Although Berlusconi is no longer in the Senate, because of his conviction for tax fraud, he still leads Forza Italia.