Scouts from both clubs have twice watched the ex-Grimsby defender, 20, who cost Posh a record £400,000 in January.
Meanwhile Sunderland want to sign Peterborough keeper Joe Lewis - but will need to offer £2.5million to get him.
A Posh source said: "We are ready to sell but the price has to be right as Joe is among the top three or four keepers in the League."
Keane, 30, is a favourite of the ex-Spurs No 2 and White Hart Lane sources were last night claiming the club were finally ready to let the Irishman go.
The source said: "Spurs would hope for at least £6million and it will be no surprise if the deal was done before deadline day."
And Hughton is set to tie up one striker deal this morning with Hatem Ben Arfa expected at St James' Park to thrash out terms over a year-long loan from Marseille.
The Magpies will have an option to buy the France winger, 23, for around £6million next summer.
Marseille president Jean-Claude Dassier said: "They offered to raise the price for the option to buy if Hatem plays 25 matches."
The Toon are also in talks with Manchester City over a loan switch for Belgian defender Dedryck Boyata, 19.
Boyata has made three appearances for City prior to this season.