Published Wednesday, 10 February 2010
Kay said he had "let down" his wife, Strictly Come Dancing presenter Tess Daly, and their two young daughters.
The All Star Family Fortunes host told The Sun: "Tess is extremely upset about this and we're working through it."
Kay, who also presents a Saturday show on BBC Radio 1, added: "I've been an idiot and I've let my family down."
Kay said the text exchanges with friends or work colleagues would start off "pretty innocent" but developed into something more explicit.
The Beat The Star presenter denied having sex with anyone he had texted but admitted he had flirted while using the VOIP service Skype which allows users to see each other.
Kay, 35, told the newspaper he treated the messages as "harmless banter" but now realised his behaviour was not appropriate for a married man.
He pledged to halt his text habit and to do "whatever it takes to resolve the situation" with his wife.
Kay added: "I've got the most beautiful family and I'm embarrassed I've put them through this."
The former model was given his first presenting role with Channel 4 youth segment T4 in 2000.
The Bolton-born presenter married Daly, also a former model, in 2003 and their daughter Phoebe was born the following year.
Their second daughter, Amber, was born in May last year.
The couple worked together in 2007 on the BBC show Just The Two Of Us which saw celebrities performing duets with professional singers.