The 29-year-old White Hart Lane skipper rejoined Spurs from Liverpool a year ago for £12m following an ill-fated six months at Anfield after his £19m summer 2008 sale.
Keane has now fallen behind Jermain Defoe and Peter Crouch in the strikers' pecking order and was rumoured to be set to go on loan to Celtic next month.
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However, reports in the Mirror suggest the Republic of Ireland captain's move might be made move permanent as Tottenham want Parkhead keeper Artur Boruc.
The Londoners could make the Boruc move an integral part of the deal as they look to replace injured keeper Carlo Cudicini who had a serious motorbike accident last month.
Keane's departure has been hastened by his involvement in arranging the squad's unauthorised Christmas party in Dublin resulting in heavy fines from manager Harry Redknapp.
Boruc, also 29, has previously been linked with moves to Arsenal and Aston Villa. He is currently contracted to Celtic until 2011.
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