The UK banks are currently owned by National Australia Bank, but NAB is expected to launch a sale shortly, the Sunday Times reported.
NAB recently pulled out of the auction of 318 branches being sold by bailed-out Royal Bank of Scotland forced by European competition officials.
With Spanish giant Santander - the owner of Alliance & Leicester, Abbey and Bradford & Bingley - now the clear favourite to buy the RBS branches, NAB is focusing elsewhere and said to be considering a complete withdrawal from the UK market.
NAB - which bought the Clydesdale in 1987 - has considered listing the UK arm, although a sale is said to be its preferred option.
BBVA has held initial discussions with NAB over the business although it has yet to make a formal approach, the newspaper said. The Spanish bank was unavailable for comment.
NAB has around 2.7 million UK customers through Glasgow-based Clydesdale Bank, which incorporates the Yorkshire Bank brand.
The business has a total of 340 branches, consisting of 188 Yorkshire Bank and 152 Clydesdale sites. It employs more than 8,200 full-time workers in the UK, with 2,400 in its Glasgow and Clydebank operations and 1,900 in Yorkshire.
Clydesdale traces its history back to 1838, while the Yorkshire was founded in 1859 in Halifax, West Yorkshire. The Yorkshire Bank joined the group in 1990.
According to NAB's first-half results last month, the UK operation boosted underlying profits to 264 million for the six months to March 31.
The bank said bad and doubtful debts were up 9% or 15 million on a year earlier but down 28% on the previous half-year.