The Hollywood actor Sean Penn has called for a billionaire to come forward and buy two squadrons of F-15 or F-16 aircraft for Ukraine in an unlikely attempt to tip the scales against the Russian invaders in the five-week-old war.

The actor’s plea for somebody to come and spend $300m on “12 aircraft with better tech than Russian MiGs or SU’s” mirrored a request a couple of hours earlier from Ukraine’s air force for US-made fighters.

Although the role of arms middleman may be one to which Penn is not obviously suited, he has been engaged in filming a documentary about Ukraine and its charismatic president, Volodymyr Zelenksiy, for months.

The Oscar-winning star of Milk and Mystic River was in Kyiv on the day the war broke out, pictured in the front row of a press briefing in the capital city, and met a tired-looking president later that day, according to an Instagram story released by Zelenskiy.

Penn’s documentary team remained for some while after in Kyiv, working closely with government officials, and the actor appears to have come close enough to Zelenskiy and his circle to try to help source higher specification arms for Kyiv’s embattled forces.

Filming for the Vice Media documentary has been going for some time, with Penn donning military gear in a visit to the Donbas frontlines in November – but Russian invasion on 24 February and Zelenskiy’s sudden global fame has made it a hot media property.

Last week, Penn threatened to “smelt” his Oscar statuettes “in public” if Zelenskiy was not allowed to speak at last week’s Academy Awards ceremony – but the Ukrainian president, a former actor, did not get a platform to address the event.

Ukraine has been asking for more fighters since the war started, but the US refused to support a deal that could have seen Poland supply 28 MiG-29s, a Russian-made jet of the type its remaining pilots are used to flying. Washington was concerned that supplying fighter jets could be deemed escalatory by a nuclear-armed Russia.

Despite the refusals, Ukraine continues to press its case, and overnight its air force tweeted “the greatest need is for fighter jets” and asked specifically for F-15s and F-16s, whose original designs date back to the 1960s and 1970s, which it said its pilots could learn to fly with “just 2-3 weeks of training”.

Experts said it would normally take pilots up to four months to learn how to fly a new fighter, and Ukraine would also have the problem of finding spare parts and engineers who knew how to maintain Boeing and Lockheed Martin-made aircraft.

Ukraine’s air force – a 10th the size of Russia’s – had been expected to be knocked out of the skies within a few days of the invasion. But some of its fighters remain active and the airspace over the country remains sufficiently contested that Russia has not been able to carry out bombing missions freely.

Andy Netherwood, a former RAF pilot and defence commentator, said: “If you’d have asked me at the beginning of the war, I’d have said it was impossible for Ukrainian pilots to learn how to master a new fighter in two to three weeks. But now I’m reluctant to use the word impossible when it comes to the Ukrainian air force.”

However, a more serious obstacle to a billionaire buying F-15s and F-16s and flying them to Ukraine would be US export controls – major arms sales have to be approved by the US Congress before being allowed to proceed. “Typically fighter aircraft are sold by one government to another,” Netherwood added.