Henry could be in charge for Aston Villa's forthcoming fixture against Swansea City, a former club of the Celtic manager Rodgers.
The newspaper reports that, after making tentative enquiries, Villa were told to make an 'all-or-nothing approach' if they wanted to prise the Ulsterman away from Celtic, his boyhood club - although it adds the Scottish champions do not anticipate this to happen.
It is also claimed that Rodgers had 'no desire' to be included on Aston Villa's shortlist of candidates to succeed the sacked Steve Bruce.
The 45-year-old is said to earn £2.5 million a year at Celtic Park, and it would have cost Villa a similar amount to free him from his contract, according to the Sun.
And while the report fails to state how much it would require to remove Henry from his current role as Belgium's assistant manager, it is unlikely to be anywhere near as much.
Meanwhile, Henry's manager, Roberto Martinez, is the latest name linked with the Aston Villa vacancy.
Is the Frenchman a better bet for Villa than Celtic's Brendan Rodgers?
Have something to tell us about this article?