L'Équipe reports that CSKA Moscow and Krasnodar are both in contact with the 30-year-old and looking to complete a deal, while Málaga and an unnamed second Spanish club are also in talks.
FootMercato says Martial has already turned down Al-Ittihad Tripoli's offer, worth a net salary of £8.6 million over two years plus a £4.3 million signing-on fee, and is now closing in on a move to Málaga.
Martial has been a free agent since leaving Liga MX side Monterrey midway through a two-year contract earlier this summer. He is looking for his third club since leaving Manchester United two years ago after nine seasons at Old Trafford, having spent one season at Greek side AEK Athens in between.
A return to Ligue 1 already looks unlikely. Auxerre made an approach earlier in the summer, which Martial turned down, and remain the only French club to have tabled a formal offer, despite the striker's representatives reportedly making his availability known widely across the division.
Lorient sporting director Laurent Koscielny, Martial's former France team-mate, confirmed the club had been offered his signature, telling Ouest-France: “I think we received 400 to 450 player offers during the transfer window. There were, for example, a lot of former internationals in the lot, including Anthony Martial.”
Koscielny and Martial won 13 caps together for France. Several Premier League clubs are also said to have been offered the forward, including Everton, who were linked after their deadline-day move for Folarin Balogun collapsed, though nothing came of the interest there either.
Málaga, who are 18th in La Liga and without a win in five games this season, appear to be the most advanced of his suitors, with reports in Spain suggesting a deal could be signed within days. Martial is expected to make a decision on his next club during the international break.