The Reds came close to landing the France international in the summer of 2018.
Fekir has gone on to become a World Cup winner while remaining a talismanic presence at Lyon, and further interest is expected to be shown in the talented 25-year-old.
Lawrenson believes Jurgen Klopp should be back in the mix, with there few areas that Liverpool can improve on during the next transfer window.
Quizzed on where the Reds could add, the former defender told BBC Sport: “When you look at this season, you can praise both of the top two teams together. One of them [Manchester City] has edged an unbelievably good title race, rather than the other failing and losing it somewhere.
“That is why it is difficult to look through the Liverpool team and say they need to sign this or that to make sure it is a different story next year.
“They will need another left-back as back-up to Andy Robertson because Alberto Moreno is leaving in the summer, and Klopp cannot expect James Milner to play there regularly.
“But it is not like the last couple of summers when it has been easy to identify what this team needs to improve it.
“In 2017 it was a centre-half, and Virgil van Dijk arrived the following January, while in 2018 it was a goalkeeper, and Alisson was the answer.
“Those problems have been solved, so primarily you are looking at adding depth in a few positions - at centre-half, or up front.
“You might argue that they don't really need any one position desperately, but I think it would be nice if they could get an attacking midfield player, of the sort they have not had since Philippe Coutinho left for Barcelona in January 2018.
“A criticism I heard a lot, particularly earlier in the season, was that with Alex Oxlade-Chamberlain sidelined, all of Liverpool's midfielders were too similar, and not good enough at opening up defences.
“I am talking about bringing in someone with a bit more creativity, like Lyon's Nabil Fekir, who they were going to buy last summer until the move fell through.
“The way the season has gone, you could not say they have missed him, but in future, they will have to look at having more ways to win games, especially against teams who just sit in.”
Liverpool fell a point short of fulfilling their Premier League title dream in 2018-19, but can still finish the season with major silverware as their attention shifts to a Champions League final date with Tottenham.