Earlier this week, the 30-year-old admitted that he would prefer to stay with the French side rather than return to Anfield at the end of his loan spell.
Garcia has admitted the club would like to retain Cole's services, but refused to reveal whether wages are proving to be a stumbling block to any proposed deal.
"It proves he is happy with us," the Lille coach told reporters on Friday. "We are happy with him.
"There has been no debate on the reciprocal desires. But this is not the time [to talk]. We have a big objective, so we don't waste our energy working on it [a permanent deal for Cole].
"What is certain is that everyone is under contract. There are special cases, and Joe is one because he is here on loan."
Since joining Lille, the former Chelsea midfielder has made 45 appearances and scored five times.