Bayrou explains resignation
Speaking on Wednesday evening, Bayrou denied that there had been any illegal or immoral practices in his party.
The accusations were aimed at undermining his position as justice minister, charged with putting the clean-government bill through parliament, he claimed.
"We can't go on living in a society of continuous denunciation," he said.
He explained that he was resigning because his position could lead to claims that he was using it to inflence the inquiry.
He praised President Emmanuel Macron and said he needed broad support in parliament, including that of Modem.
On Tuesday Defence Minister Sylvie Goulard, who was a member of the European parliament from 2009 to May this year, said she could not remain in the government while facing possible investigation.
Goulard's and Bayrou's party, Modem, faces allegations it used European parliamentary funds to pay party workers actually based in France.
Bayrou on Tuesday declared he would not be leaving but by Wednesday morning he had changed his mind and said he would not be in the new cabinet, due to be unveiled later in the day.
European Affairs Minister Marielle de Sarnez, the only other Modem member in the government, also announced she was standing down to head the MoDem parliamentary group in the National Assembly.
Key Macron ally in elections
Modem, a small, centrist party, brought key support to Macron in the presidential and parliamentary elections.
But the president's party, Republic on the Move (LREM), won enough seats to rule alone in the legislative poll.
As justice minister, Bayrou was responsible for a clean-government bill that is a key part of Macron's promise to transform French political life.
But not only is he a possible subject of the jobs inquiry, he also raised eyebrows by phoning Radio France journalists to press them to go easy on an investigation into the question.
Nevertheless, he said his resignation was not dictated by anyone else, presumably a reference to Macron.
Government spokesman Christophe Castaner said the decision was personal, because he "wanted to defend himself in this affair", but also commented that it "simplifies the situation".
Right-wingers lay into Macron
Interior Minister, Richard Ferrand, a former Socialist, has been hit by a separate scandal and Macron has asked him to quit the government to lead LREM's parliamentary group.
After Bayrou's announcement, Laurent Wauquiez, of the mainstream-right Republicans, called the resignations a "political scandal" and a "major crisis" coming less than a month after Macron took over as president.
Florian Philippot, whose far-right National Front also faces an inquiry into alleged misuse of European funding, mocked Macron's promise to bring morality to the country's political life and called on Labour Minister Muriel Pénicaud, who may be implicated in another legal inquiry, to resign as well.