As reported by The Daily Telegraph, the Tigers' bid is a late attempt to swoop on the 29-year-old lock, who is also gaining attention from the Dragons, Cowboys, Sharks and Titans.
Each club is understood to have lodged an offer to Finucane's party, who are hopeful a keen suitor will bump the length of their tabled deal to four years.
While clubs look reluctant to do so, the Tigers have suddenly re-entered the race for the New South Wales forward, while Newcastle look to have pulled their two-year offering, according to News Corp's report.
It is also understood that Melbourne have come to terms that their premiership lock won't be at the club next season, leaving just the five clubs to be in strong consideration for Finucane's services.
Finucane's final destination could be the result of a busy transfer market from the past week, with the Knights sewing up a significant amount of cap room with the imminent signing of Dane Gagai.
The Tigers' sprung interest could also be a result of their failed attempt to lure out-of-favour Broncos forward Tevita Pangai Junior to Concord.
The second-rower is set for a short-term venture with Penrith before heading to Canterbury from 2022.
“We are definitely building through our youths and looking for great talent to continually join us. Sometimes you get them and sometimes you don’t but Tevita is a good fellow and I wish him all the best.”