Blues GM Doug Armstrong has been following the Flyers for a good portion of February, catching them in Toronto on Feb. 11, and having attended two of the Flyers' last three home games.
Briere, 35, owns a no-movement clause and would have to approve any deal the Flyers try to complete.
The Flyers have not asked Briere to waive his NMC and the Daily News believes the team is not interested in moving him.
"I don't want to comment on anything at this point," Briere told paper. "Trades are not my department. I've been a Flyer for a long time. My heart belongs in Philadelphia with the Flyers. I can't see myself playing anywhere else."
The Blues have been looking to acquire a top-six forward for some time, and Armstrong's search might have been increased with the season-ending loss of Jamie Langenbrunner and a recent concussion to rookie Vladimir Tarasenko.
Briere, who has an annual salary cap hit of $6.5 million, has two years left on his contract beyond this season that will see him earn $3 million next season and $2 million in 2014-15.
According to the Daily News, the Flyers are interested in Blues defenseman Kevin Shattenkirk, but Armstrong is not interested in moving the 24-year-old.
The Bruins, meanwhile, have also asked the Ottawa Senators about captain Daniel Alfredsson's availability, which is non-existent, and GM Peter Chiarelli continues to scour the market.