Sovereign wealth funds, French insurers, big companies and wealthy families are among the first investors to commit money to the new fund, which is being called Silverlake.
"The aim is to eventually raise a bit more than 10 billion euros when the (capital) raising is done," Bpifrance Chief Executive Nicolas Dufourcq said.
"We're doing a first closing that should be several billion," Dufourcq told a journalists, declining to give names of the initial investors.
The new fund would give extra financial firepower to Bpifrance, which already manages a 15 billion euro portfolio of state shareholdings.
Its investments include some of France's biggest companies such as carmaker PSA, chapmaker STMicrolectronics, telecoms group Orange and car parts maker Valeo.
Silverlake would bring in outside capital that could be used to help shore up the equity structure of French companies destabilised by activist investor campaigns or in the midst of a tricky business or shareholder transitions requiring a long-term investor commitment.
"France doesn't have any pension funds, so we're doing the job," Dufourcq said.
The fund, which is to be formerly presented by Finance Minister Bruno Le Maire in the coming weeks, aims to make about 15 investments with a time horizon of about 10 years and offer investors dividends for putting in their capital.
While Bpifrance builds up Silverlake, it would also continue selling down assets to optimise its portfolio, Dufourcq said.
"We've got a pretty signficant divestment programme this year, at least as big as in 2019," when Bpifrance sold 1.6 billion euros (1.35 billion pounds) worth of assets in its portfolio, Dufourcq said.
In addition to managing French state shareholdings, Bpifrance's main business lines also include offering guarantees for banks' business loans and export credit insurance.
It aims to launch a new retail savings product this year around a fund invested in other funds with interests in some 2,500 French companies.
(Reporting by Gwenaelle Barzic; Writing by Leigh Thomas; Editing by Alison Williams)