The Rams hold the selection for the second time in two years, as in 2012 they traded with the Washington Redskins to land their sixth and 39th selections that year along with their first-round picks in 2013 and 2014.
Washington landed Robert Griffin III with that pick, but Fisher believes that the depth of the Draft will prevent teams moving up this time around.
"I think it's highly unlikely that even if we did trade out that would we get the value that we did the first time around," Fisher told ESPN.
"This Draft is unique in that I think you're going to get players in the middle part of the third round that potentially could start, and start very early for you.
"It really depends on the positions, but there will be a lot of teams excited about who they have acquired in the first couple rounds because it's such a deep Draft."
The Rams finished with a 7-9 record in the 2013 season.