The 24-year-old Manchester United striker and his television presenter wife have faced intense scrutiny since a tabloid newspaper ran a story about their private life last Sunday.
"The last six days have been extremely painful for us and for our families," the couple said in a statement. "It is impossible for us, as it would be for any family, to attempt to resolve any issues in the current media glare and against the backdrop of so many inaccurate and intrusive stories.
"We would therefore ask that the media now respects our privacy and the right of our family to discuss these matters in private."
Rooney and his wife, also 24, met when they were still at school and married in June 2008. They have a 10-month-old boy called Kai.
Rooney, who joined Manchester United in 2004 from Everton, scored in an international match against Switzerland on Tuesday and is due to play against his old club on Saturday.
Manchester United manager Alex Ferguson told a news conference on Friday that he would not discuss the player's private life.