Robben, 34, has confirmed this campaign will be the last of his highly successful spell at Bayern, whom he joined from Real Madrid in 2009.
The ex-Netherlands international is yet to announce his next destination, but hometown team Groningen are believed to be interested in engineering a move.
PSV, where Robben spent two seasons over a decade ago, are set to enter the race after the club's general manager Toon Gerbrands revealed Van Bommel has been in touch with his former team-mate.
"There has been contact. Mark, who played with Arjen, sent him a message," Gerbrands told Omroep Brabant.
"It is, of course, up to Arjen to make the decision."
The seven-time Bundesliga winner first linked up with Van Bommel when he moved from Groningen to PSV as a teenager.
The pair were later reunited at Bayern and spent several years together in the Netherlands national team.
Robben has appeared 15 times in all competitions this term, scoring five goals.
He is one of a number of senior Bayern players that is set to move on in the new year, with board members at the club indicating that the playing squad is to be freshened up in the near future.
The German champions are currently languishing in third place in the Bundesliga table, nine points behind leaders Borussia Dortmund.
Niko Kovac, who was appointed as manager of the Bavarians last summer, has endured a difficult start to his time as Bayern boss, but CEO Karl-Heinz Rummenigge and president Uli Hoeness have insisted that the former Eintracht Frankfurt coach retains their full support as it stands.
It's been suggested that Franck Ribery, Rafinha and Jerome Boateng are among those that could also be shown the exit door at Bayern in 2019.