The pair have known each other for over a decade since meeting at the Sanchez-Casal Academy in Spain.
Murray opted to make a number of changes to his backroom staff, hiring Amelie Mauresmo as coach to replace Ivan Lendl after he left his post in March, before splitting with Vallverdu and fitness trainer Jez Green.
The departure of Venezuelan Vallverdu, who is now coaching Tomas Berdych, was unexpected as he had become increasingly important under Lendl.
Murray had hinted at divisions in the camp in an interview with The Independent earlier this month but Vallverdu is adamant there is no problem between them.
“We were all pulling in the same direction,” he told the Daily Mail.??”I’m quite proud of how we pulled it around after the US Open. It was pretty hard, him qualifying for (the ATP Tour Finals in) London from his position after New York.
“Of course Andy is the one who played the matches but everyone had the goal of helping him get there. I don’t think the last four months are a reason we aren’t working together any more, it was genuinely mutual.
“His determination, professionalism and stubbornness, that’s what has made him so successful but, because of the way he is, it gets to the point when it is time for both parties to go their own ways.
“Andy and I have been friends for nearly 15 years. Our personal relationship is still good, I consider him one of my best friends and I’ll always be in his corner, except for when I am coaching against him.”