Both Simon Mignolet and Loris Karius have failed to convince between the sticks at Anfield, with the former usually preferred by Klopp in the Premier League and the latter in the Champions League.
Alisson, 25, has played every single minute of Roma's Serie A campaign thus far, conceding just 12 times in 18 matches. He has more than three years left on his contract at the Italian club.
The Brazil international served as back-up to Wojciech Szczesny last season after arriving in the Italian capital from Internacional. Szczesny joined Juventus last summer, allowing Alisson to take over as first-team goalkeeper at the Stadio Olimpico.
"Taking stock  was a very positive year for me at Roma and with the national team," Alisson was reported as saying on 3 January.
"I took the number 1 shirt which is what I wanted, to be the starter in every match. It was a special year for me.
"I was waiting for this year, last season I didn't play much but I worked so hard to get myself ready. When Roma bought me I was ready, I'd have played well, even right away.
"I had faith and I want to keep doing well for Roma."
Liverpool have also been linked with a move for Stoke goalkeeper Jack Butland, who is in contention to be England's first-choice goalkeeper at the World Cup in Russia.
Meanwhile, Van Dijk is eligible to make his Reds debut in the FA Cup third-round tie against Everton at Anfield after he completed his world-record £75m move from Southampton earlier this week.