The former Hearts centre-back is on trial with Rangers while McCoist has also declared an interest in Scotland international midfielder Don Cowie, who is set to leave Cardiff.
And the Rangers boss admitted he was "really keen" to sign the Lithuanian.
McCoist told Sky Sports News: "Marius has come up after being released by Leeds United and trained with us and we are going to have a chat.
"Obviously we'll have a look at the budget, I think there's still a few quid knocking about in that budget and that will be put to the test in the next week or so."
McCoist, who has added Darren McGregor, Kenny Miller and Kris Boyd to his squad ahead of their Scottish Championship title bid, added: "It's our job to strengthen as much as we can.
"We have still got a bit in the budget and we want to do that because it's vitally important that the team get back to the top flight."