The Lions are in the market for a defender after allowing Scott Barron to join Brentford earlier this week and Painter, who is third-choice at Albion this year, could be available.
The Midhurst and Petworth Observer also claims boss Kenny Jackett has seen a £300,000 bid rejected for Barnsley captain Jim McNulty – though the Millwall boss has since denied that.
And Jackett has played down suggestions that is looking to fill the void left by Barron's departure:
“We have Malone, we have Lowry, who is capable of that position, and we have Jack Smith," he said.
“We don’t have a big squad and I think there are other areas that need strengthening more than there.”