Taylor has attracted a lot of attention from Premier League clubs and last week reportedly had a medical at Reading.
But the move was put on hold while Harry Redknapp searches for a replacement.
The Pompey boss said: "Matty wants to leave. He wants to move and we have had a couple of clubs in for him.
"But I can't sell anyone until we get some people in."
Middlesbrough may have a fight on their hands to persuade Taylor to move north as he is believed to be reluctant to move his family from their Oxfordshire home.