Marussia is offering to make Swiss Leimer, 25, its test and reserve driver with immediate effect.
"He could enter right now as a test driver for Marussia," his manager, Sven-Oliver Mangold, confirmed.
"It has even been promised that next year he will be one of the regular drivers," Mangold added.
He said it will cost Leimer a sponsorship figure in the "low single-digit million range".
Therein lies the problem. Until now, Leimer's career has been bankrolled by the wealthy Rainer Gantenbein, who over the years has ploughed over EUR 15 million into the burgeoning driver.
But Gantenbein said some months ago that he is "no longer willing" to burn money on the F1 dream, because the sport's system of pay-drivers is "sick".
"It's a bottomless pit," he insisted. "At some point you have to pull the plug."
Leimer's manager Mangold, therefore, is scrabbling to raise the necessary money before other opportunities also start to slip away.
He said talks with a "prominent Swiss company" have fallen through.
"I always get the same answer: we only sponsor teams, not individual athletes," said Mangold.
In the meantime, the clock is ticking.
"Marussia has suggested (Leimer contest) the test at Silverstone and then to target the next race," said Mangold, referring to the German grand prix next month.
And what if that deadline passes? "Then formula E would be an alternative," he said, amid reports many well-known names are rescuing and rebuilding their careers in the FIA's new all-electric single seater series.