Roberts, who was named HSBC Player of the Tour when they Lions travelled to South Africa three years ago, has been heavily linked with a switch to France to join big spenders Racing Metro next season, although he has yet to make a formal announcement on his future.
"The offer that we made to Jamie was an extremely attractive one and would have seen him become certainly the highest paid player in the Blues squad, and arguably in the UK," said Blues CEO Richard Holland.
"As a business we simply don't have the resource to compete with the offers that Jamie has received."
If, as expected, Roberts does opt to head to France next term, he will join a whole host of Welsh stars plying their trade in the Top 14.
Fellow Lions Mike Phillips, James Hook, Lee Byrne and Gethin Jenkins are at Bayonne, Perpignan, Clermont Auvergne and Toulon respectively, while internationals Luke Charteris, Aled Brew and Huw Bennett also left home to cross the English Channel in recent months.
Wales caretaker coach Rob Howley issued a stark warning to his French-based stars when he picked Tavis Knoyle ahead Lions favourite Phillips for Saturday's autumn opener with Argentina but the warning appears to have fallen on deaf ears as far as Roberts is concerned.
Blues chairman Peter Thomas believes the growing number of big names opting to leave Welsh shores will have serious consequences for both the regional and national sides and he has called on the Welsh Rugby Union to act quickly to stop a further exodus.
"It is impossible for the Welsh regions to compete with the kind of money that is on offer from other clubs in other countries," said Thomas.
"The time has come that the Welsh benefactors should not be responsible for funding the professional game.
"The chairman of the regions can guide companies but if we want to keep Jamie Roberts and other international players in Wales then the governing body has to intervene.
"The player drain out of Wales, which has a negative impact on the Welsh regions and the national team, will only continue without increased funding from the governing body.
"We have invested a lot of time, money and effort developing Jamie as a rugby player from an early age after identifying him as a star of the future.
"We have stuck by him through thick and thin and supported him during last season when he only completed two matches for the Blues because of his knee injury and international duties, playing 15 matches for Wales."
