The 23-year-old has played just 19 minutes of Premier League football for the Clarets this season, making substitute appearances in contests against Sheffield United and Arsenal.
Obafemi only returned to first-team action in the middle of November after surgery on a hamstring injury sustained whilst on international duty with the Republic of Ireland in June.
The forward spent the second half of the 2022-23 term on loan at Turf Moor from Swansea City before making the switch to Lancashire permanent in the summer, scoring two goals in the Championship last campaign as Vincent Kompany's men eased to the second-tier crown.
According to transfer expert Fabrizio Romano, Obafemi is closing in on a move to Millwall for the second half of the 2023-24 season.
The report claims that the Irishman has a medical booked at the Lions on Tuesday, and assuming all is well, the attacker will be welcomed into Joe Edwards's ranks.
It is understood that Burnley and Millwall have reached an agreement in principle over the signing of Obafemi, meaning an announcement could be made before the weekend once the medical is passed.
Looking for extra firepower up front, Kompany's side secured the services of David Datro Fofana on loan from Mauricio Pochettino's Chelsea on January 13.
The arrival of the Ivorian has seen Obafemi fall down the pecking order even more in attacking areas, with the likes of Zeki Amdouni, Lyle Foster and Jay Rodriguez already preferred by Kompany.
Following the sacking of Gary Rowett and hiring of head coach Edwards earlier in the campaign, Millwall have
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