The Mirror says Liverpool and Tottenham had reached terms over an outline of a deal for the Scotland fullback.
However, the move has now collapsed after Liverpool's defeat at Bournemouth.
It's suggested with Roma refusing to release on-loan left-back Kostas Tsimikas back to Liverpool, it's been decided Robertson will not be allowed to leave this week.
The original agreement would have seen Robertson leave for Spurs after this week's Champions League round.
Robertson is off contract at the end of this season.
Meanwhile, Roma chief Ricky Massara said on Sunday: “They are all situations that are evolving, we are in a dialogue with Liverpool and we’ll see how that evolves over the next few days.
"We have a week left to see if there are any solutions that can truly strengthen the squad, which is already competitive.”