However, Traore is now thought to be their top target as they look to beat the likes of Real Madrid, Barcelona and Manchester United to his signature.
Here, Sports Mole looks at what shirt numbers could be available to the Spaniard at Anfield.
Traore's choice of shirt number could be very straightforward at Anfield; he currently wears the number 37 shirt for Wolves, and that has been vacant at Liverpool since Martin Skrtel left the club in 2016.
Considering he also wore it at Middlesbrough he may opt to stick with that on Merseyside, although he has also previously worn the '20' for Aston Villa.
That shirt is currently taken by Adam Lallana, but the midfielder is expected to leave Liverpool when his contract expires this summer and so the '20' is likely to be available ahead of the 2020-21 campaign.
Traore previously wore '27' and '28' during his brief Barcelona career too, and while the first of those is taken by Divock Origi, the '28' is free and available to be claimed.
Of course, the speedy winger could opt to choose a new number which would perhaps reflect his importance to the team more accurately, although vacant options are few and far between for Liverpool right now.
The most commonly-associated ones for wingers are '7' and '11', which are occupied by James Milner and Salah at the moment, and the lowest number which is not currently taken is the '16'.