The club's new coach Paul Roos has discussed Martin's future with the 22-year-old's manager Ralph Carr as the club attempts to bolster its midfield.
Roos has previously indicated he would be prepared to trade the club's No.2 pick if the right player was available, in order to add talent immediately to Melbourne's list.
Demons' football manager Josh Mahoney said the club had made initial contact with Martin's management.
"Melbourne did make enquiries to his management about his intentions for next year," Mahoney said. "We are waiting to hear back from them before we discuss any next steps we might take as a club."
Richmond shocked football watchers on Monday when it released a statement indicating that Martin wanted to test the market and was open to offers from other clubs.
The Tigers have been in negotiations with the midfielder for several months but have been unable to secure his signature. Richmond reached the finals this year for the first time since 2001 and looks to be building an exciting squad for the future.
The club remains hopeful it may still retain Martin but it is unwilling to compromise its player payments strategy to keep one player, regardless of his talent.
Richmond is looking to establish a flatter payments structure similar that employed by recently successful clubs.
Football manager Dan Richardson made it clear Martin remained a required player.
"We are proud of the way Dustin has developed as a footballer, and as a person, in his time at Richmond, and we have been steadfast in our view that he is a required player who has an exciting football career ahead of him," Richardson said.
Melbourne is keen to build both talent and depth through its midfield. It averaged just 40 inside 50s per game in 2013 and finished with a percentage of 54.
Roos, the new Demons coach, is also scheduled to fly to Perth on Tuesday to meet with 18-year-old forward Jesse Hogan, his family and management team to outlines his plans as coach.
