Defenders Robert Huth and Jonathan Woodgate agreed to join EPL side Middlesbrough.
Spanish newspapers said Real were already in negotiations with Newcastle United over the defender.
The 25-year-old defender will remain in the Spanish capital for most of June in an attempt to regain full fitness.
The 25-year-old England international tore the hamstring in his left leg when playing against Zaragoza
Goalkeeper David James is also set for a surprise return to the England squad for the World Cup warm-up match against Argentina.
The England defender's calamitous start to his Real Madrid career continued as he scored his second own goal in three outings for the Primera Liga side.
Real Madrid's Jonathan Woodgate has confirmed he will play on loan in England, but the team hasn't been confirmed.
An 86th-minute header by Jonathan Woodgate gave struggling Spurs a much-needed win and continued Hull's steady slide towards the relegation zone.
England coach Steve McClaren is hoping defenders Micah Richards and Jonathan Woodgate will shrug off injuries in time for their Euro 2008 qualifiers against Israel and Andorra.
Tottenham Hotspur defender Jonathan Woodgate scored the winning goal in extra-time as the north London upset the defending champions.
The club is planning to strengthen it's bid to return to the Champions League by signing two players in next month's transfer window.
The Daily Express said the 31-year-old former England captain had signed "the most mind-boggling contract in global sport".
10-man Real Madrid fought back from a goal down to claim a 3-1 win at home to Athletic Bilbao.
Leeds United returned to the Premier League after a 16-year absence on Friday as West Bromwich Albion's loss at Huddersfield Town guaranteed them a top-two finish in the Championship. Three times English champions, Leeds were relegated from the Premier League in 2004 after hitting financial trouble and being forced to sell a host of big-name players such as Robbie Keane, Jonathan Woodgate and Lee Bowyer.