The former England Under-21 international rejected a four-year offer worth £45k-a-week, and the Magpies have put Taylor up for sale as a result.
Taylor, who is worth around £4 million has been in talks to get a new deal with Newcastle for months, with his current contract set to expire next summer.
However, with contract negotiations stalling, Everton and Blackburn are now interested in bidding for Taylor, according to The Mirror.
The relationship between Taylor and Newcastle soured after sources leaked that Taylor would not be receiving a new deal and that Toon intends to sell the defender.
Reports suggest that the 24-year-old wanted to be paid £60k-a-week in order to be on a similar salary to team-mates Alan Smith, Joey Barton and Xisco.
Taylor is currently out injured, recovering from a dislocated shoulder, but the Englishman is targeting a return in about a month.
Newcastle manager Chris Hughton insists that he wants to keep Taylor, a stance that is opposite of what the club announced after failing to reach an agreement on a new deal.