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Published 15:35 4 May 2016 BST
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The news was all-but confirmed by Tony Mowbray, manager of Cole's current club Coventry City.
The England international midfielder has not turned out for his country since a substitute appearance in the 4-1 defeat to Germany at the 2010 World Cup.
Now 34, he managed one cup appearance for Aston Villa before joining Coventry, initially on loan.
Adu, meanwhile, is with his 12th professional club (13th if you count KuFu-98, the reserve team of Finnish outfit KuPs, with whom he spent some time on loan).
"Joe's going to join Tampa Bay and it might happen pretty quickly if he can get international clearance etc," Mowbray told the Coventry Telegraph.
"He's indicated to me that when their season finishes in October that he might come back here because there's a window of opportunity for him to come back on loan.
"I said that would be great but it will be dependent on how many loans we have got at that time, of course, so let’s see what happens."
Cole has played 22 games for Coventry, scoring twice as the club flirted with promotion before falling away towards the end of the current campaign.

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