James Beattie
Hodgson is hoping to bring in an experienced head who can add something different in attack - and believes Beattie fits the bill.
The powerful target man, 32, played under Hodgson at Blackburn Rovers in the late 1990s after coming through the club's youth system.
And Hodgson highly rates the player, believing he will also represent value for money this summer because of his age and his well-publicised bust-up with Stoke City manager Tony Pulis last season.
Beattie is not seen as a long-term option, but the new Reds manager is thought to be prepared to risk the ire of the Anfield faithful by signing a player he believes will have an instant impact on the squad.
The Potters are thought to be keen to cash in on the striker, especially after reports Pulis has left the £3million signing out of his side's pre-season tour of Austria.
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"I managed Joe before, and I signed him when he was 11," Redknapp told ESPN Soccernet.
"We have always had a good relationship. I know what makes Joe tick and it's his love of the game, and his need to be playing it. That is what I am offering him.
"I have spoken with Joe at length and I would like him and, when we talked, he appreciated my view that Tottenham is now a club going places.
"I would like Joe here with me, but it is now up to the club and up to Joe and his agent and his dad, and our chairman, Daniel Levy, is talking to them to try to do a deal. If it is beyond us in financial terms I can appreciate that but I am hoping other factors will come into play to make up Joe's mind."
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Taiwo, who currently plies his trade in France with Marseille, would also be a replacement – for the Fiorentina-bound Emiliano Insua.
Tottenham are also interested in the £8.5 million-rated Nigerian international left-back and it would be a straight choice between the more prestigious Liverpool or Champions League side Spurs.
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Cole has had a good run. He played far more games than he missed at Chelsea and became a regular member of the England side under Sven Goran Eriksson and Steve McClaren. But he does seem to be making the mistake of thinking he's better than he actually is. Apparently, Cole would prefer to stay in London and - only a few months since he and his wife had their first child - that is understandable.
But should he decide that no English club can afford what he feels he deserves, he may yet be tempted by a move abroad, with both AC Milan and Valencia reportedly interested.