Martin's protracted contract negotiations with the Tigers reached a flashpoint this week when the 22-year-old requested more time to consider offers from other clubs.
Richmond expressed its extreme disappointment at Martin's request on Monday, but the midfielder took to Twitter later that day to deny he had walked out on the club.
AFL.com.au understands Martin was due to meet with a Giants delegation on Thursday led by senior coach Leon Cameron, list manager Stephen Silvagni and CEO David Matthews.
The Giants appear to be in the box seat to snare Martin should he decide to leave Richmond, with reports on Thursday suggesting Melbourne had pulled out of the race. having baulked at the midfielder's apparent asking price of $600,000 a season.
However, the Brisbane Lions are also believed to be interested in talking with Martin, the No.3 pick in the 2009 NAB AFL Draft.
The Tigers had been confident of retaining Martin before this week, but were unwilling to shift from their long-term player payment strategy to secure him.
"Given the considerable effort we have put into Dustin's development both on and off the field, and our efforts to retain him all season, we are extremely disappointed with this decision," Richmond football manager Dan Richardson told richmondfc.com.au on Monday.
But Martin dismissed reports his impasse with Richmond meant he had walked out on the club.
"At the weekend, I asked my manager Ralph Carr to request an extension of time from Richmond, to allow me to fully consider the options about my future," Martin tweeted on Monday.
"This decision was made purely by me, together with my family.
"As you can understand, this is a very important decision for myself and my family and one that I won't make lightly.
"I want to make it clear that I appreciate everyone's patience in allowing me the time to make this decision.
"The claims that I have walked out on the Richmond Football Club are totally incorrect.
"I can tell you that at this time, no decision has [been made]."
Martin has played 86 games and kicked 90 goals in four years with Richmond, and was third in the club best and fairest in 2011.
Twitter: @AFL_Nick
