The 30-year-old left for Elland Road in November and there were indications from Leeds that Neil Warnock was keen to keep him until the summer.
But, despite playing in 11 of the 12 matches for which he was available, Tate was not part of the 18-man squad that beat Bristol City 1-0 on Saturday and has been allowed to leave.
Warnock was effusive in his praise of Tate after he made his debut in Leeds’ 2-1 win over Crystal Palace on November 24.
Speaking after the game, he said: ‘Alan Tate is a good pro. He had a calming influence on us at the back … We needed the lift in confidence.’
The Leeds manager is understood to have considered a £275,000 bid for the defender earlier this month, but settled for extending his initial loan deal to January 26.
