Messi, 33, shocked Barca on Tuesday when he informed them of his desire to find a new home this summer after nearly 20 years at Camp Nou.
Barca supporters gathered outside the offices at the club's stadium upon learning of Messi's request and chanted for Bartomeu to resign in the hope that it would convince the Argentina forward to stay.
Jordi Farre, who plans to run for the presidency in next year's elections, has also called for Bartomeu to quit and is leading the vote of no confidence to have him removed.
"We consider that Messi's request to leave Barcelona is the latest sign of the mess the club finds itself in," a statement from Nou Impuls Barca, which is the slogan for Farre's campaign, said.
Farre will have 14 working days to gather the signatures of 15% of the club's members, which is just under 17,000 people.
If he manages that, the vote of no confidence would then be put to the entire membership base in the form of a referendum. Two-thirds of a potential 154,000 voters would have to go against the board in the second stage in order for an election to be called immediately.
In July, another group of Barcelona members had announced plans to launch a vote of no confidence against Bartomeu because of the "unacceptable management" of the club.
In a statement released before the humiliating defeat to Bayern Munich in the Champions League, Cor Blaugrana, the collective of Barca fans from various supporters' groups behind the action, said they had grown fed up with the Catalan club's top brass and argued that change was needed immediately.
However, unlike Farre, they have not yet filed the motion and may instead decide to divert their attention towards supporting Nou Impuls Barca's movement.
Barca will hold elections in March 2021 to elect a new president regardless of what happens with Farre's motion. Club statutes dictate that Bartomeu cannot stand again.
Following Messi's decision on Tuesday, Farre's fellow presidential candidates, Victor Font and Joan Laporta, also joined the chorus of calls for Bartomeu to resign.
Messi, who has a contract until 2021 and a €700 million release clause, has grown unhappy at performances on the pitch -- especially in Europe -- in recent seasons and has also been frustrated by mismanagement at the top level.
He clashed with former sporting director Eric Abidal earlier this season and was critical of the club for leaking information about negotiations over a wage cut at the start of the coronavirus pandemic to "exert pressure on the players."
Manchester City are among the clubs keen to sign Messi. They are crunching the numbers to work out if it's financially possible, while sources have told ESPN Messi has already spoken with City coach Pep Guardiola, who he previously played under at Barca.
Messi hopes a clause in his contract will allow him to leave for free but Barca insist that clause expired in June.