The 30-year-old spent the second half of last season on loan at West Ham United, who were relegated from the Premier League.
And despite previously admitting he would have no problem staying at Upton Park, the Hammers now seem highly unlikely to follow up any interest due to financial constraints.
That leaves the way open for a number of Premier League sides to bid for Keane, who would be available on a free transfer in 12 months time.
Newcastle have tried twice in recent times to sign him, but some suggest he would be happy to snub a Geordie offer to stay in London, with Fulham a likely option.
Robbie Keane: ‘I’ve only got one year left there so I am sure Spurs will probably want to do business on me.
‘For me it is about enjoying my football and playing as much as I can.’