Hull are on the hunt for a goalkeeper as normal first-choice 'keeper Allan McGregor is ruled out for around five months with a back injury.
Marshall previously played in the Premier League with Cardiff City in the 2013/14 season and despite the Welsh club's relegation, the Scot made the Sky Sports Premier League Team of the Year, chosen by Gary Neville and Jamie Carragher.
The 'keeper has made 278 appearances for the Bluebirds since joining the club in 2009 and also has 22 caps for Scotland.
Should Marshall join Hull, he will join fellow Scotland internationals in Allan McGregor, Andy Robertson, Robert Snodgrass and Shaun Maloney.