L’Équipe state that the Nigerian international is close to leaving this month after his move from Sunderland in the January of 2019.
Maja jas suffered several injuries during his time with the Ligue 1 side leading to a lack of appearances.
Bordeaux have reportedly been approached by several teams over a loan move but the French club would prefer a permanent deal.
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Serie A club Venezia and Alaves of La Liga, as well as two MLS clubs, have also enquired about the player.
Maja, however, is only interested in a return to England with Bordeaux keen to wipe his €90,000-per-month salary from their wage bill.
The 23-year-old spent some time on loan in the Premier League with Fulham last season.
Maja scored 16 goals during the 2018-19 campaign with Sunderland in League One before being sold in the January of that season.
Championship club Nottingham Forest were said to have agreed a deal to pay Bordeaux a fee of €4.5m for Maja last summer.
That roughly equated to around £3.8m for the striker but the deal fell through due to an issue with Maja’s medial.
But a back issue highlighted in his medical saw that deal fall through and Maja return to France.
Maja – now a Nigeria international – has 18 months left on his deal with Bordeaux thought to be asking for at least £3million.
Having sold Maja to Bordeaux midway through the 2018-19 season, Sunderland are entitled to 10 per cent of the profit Bordeaux made on Maja.
Maja’s departure from Wearside mid-way through the campaign was a controversial episode with the player and his agent often portrayed as the villain.
Speaking about his Sunderland exit to The Echo, Maja said in 2019: “Basically, I wanted to stay. Given the right contract, I would have stayed.
“But at the time, I don’t think Sunderland were prepared to give me what I wanted so when the opportunity to play in one of Europe’s top five leagues came up, I couldn’t turn that down.
“It wasn’t an easy decision, but it was one which to improve my career and my game I think I had to make.”
“I have ambitions in my career and in my life and this is just a stepping stone for me.
“I want to keep pushing and fans need to understand what players want for themselves.
“Even though that is a bit selfish, it’s a selfish game, isn’t it?
“For me it was important to make that move and playing at a higher level, it’s a given.”
Speaking after Sunderland’s failed promotion campaign under Jack Ross, Maja added: “To see them not get promoted was really upsetting, and I wished them all the best when I left.
“But that’s just football, you have to bounce back the best you can and hopefully they do that this season.”