The powerful attacker, 24, enhanced his reputation in South Africa and would cost in the region of £20m from Rennes.
Despite missing the crucial penalty that would have taken Ghana into the semi-finals, he made the shortlist of 10 players for the coveted Golden Ball award.
New Reds boss Roy Hodgson has also been working hard to make Joe Cole his first prize capture of the summer.
Liverpool are talking to Cole and his representatives, but fear the Londoner wants to stay near the capital for family reasons.
The Reds are prepared to pay around £100k a week to Cole, who'd be guaranteed more first team football than he may get at Arsenal or even Spurs.