Unions had called for all companies to scrap dividends to preserve cash and for French state-backed groups to take a lead, a stance backed on Friday by Finance Minister Bruno Le Maire.
The government is deferring companies' tax and payroll charges and has offered to guarantee up to 300 billion euros in business loans to keep firms afloat through the crisis.
"If you do not have the cash to pay tax or social security charges, then you also don't have cash to pay your shareholders, so don't pay a dividend," Le Maire said on BFM TV.
He added that any companies that benefited from the exceptional tax relief and were found to have paid a dividend would be forced to repay the money and a penalty on top of it.
Meanwhile, Le Maire said he would not sign off on any state guarantee for a loan to a company that did not scrap its dividend.
As for companies in which the state directly holds a stake, the government's representatives would vote against paying a dividend if the company benefited from public financial aid.
President Emmanuel Macron has had a fraught relationship with unions since he came to office in 2017, pushing through reforms including a labour market overhaul which faced stiff opposition from workers' representatives.
Unions have regained some clout in France and other countries hit hard by the coronavirus outbreak, including Italy, as they push the state to better protect employees and call for scrutiny of the strategies taken by private companies.
"This government (in France) has not held unions in much esteem," a union source at one state-backed French firm said.
"But now, even if it's maybe not a case of respecting unions, they might be a little more wary of what we have to say, and might listen to us more."
If dividends are suspended, this might help unions persuade workers to stomach any negotiations further down the line on salaries for instance, the source added.
Italy's trade unions said this week the government in Rome had agreed to extend the number of companies that would temporarily close, after they threatened to strike.
France's stance on dividends from companies in which the state has a stake would impact groups ranging from EDF, which is just over 80% government owned, to carmaker Renault, which is 15% state backed.
The French state also has holdings in Air France-KLM, lottery group FDJ, telecoms group Orange and defence firm Thales, among others.
State-backed Airbus and aerospace supplier and engine maker Safran have scrapped their payouts in recent days as they grapple with the fallout of the crisis.
Others have yet to budge, however. The chairman of Engie said on Friday the utility and its board needed more time to decide.
Scrapping dividends altogether would also deprive the government of income. Foregoing the 2019 payout from Engie would mean a shortfall of some 461 million euros (407.5 million pounds).
Non-state backed companies are also wavering.
Ray-Ban maker EssilorLuxottica said on Friday it had yet to decide what to do even as it scrapped its financial targets for 2020, while advertising group Publicis has also dropped its financial guidance but maintained its dividend.
(Reporting by Henri-Pierre Andre, Sarah White, Dominique Vidalon, Bate Felix, Marine Pennetier, Benjamin Mallet and Leigh Thomas; Editing by Edmund Blair, Susan Fenton, Alexander Smith, Kirsten Donovan)
By Henri-Pierre André and Marine Pennetier