The Scot excelled in the Barclays Premier League last term but Cardiff City will demand at least £15m.
It is understood that Arsenal - who have started the search for a goalkeeper to compete with Wojciech Szczesny - may now shift their attention to Norwich City's John Ruddy. The Englishman is thought to be available for £6m.
Both players would likely consider a move after their current clubs were relegated to the Sky Bet Championship this season. Cardiff finished the campaign at the bottom of the table, although Marshall was still celebrated for his performances.
The 29-year-old cost Cardiff just £500,000 from Norwich in May 2009. He signed a new four-year contract at Cardiff in February.