Marshall, considered one of the best goalkeepers outside the Premier League, was the subject of interest from West Bromwich Albion last summer and has also been linked with Everton.
His Scotland colleague, Allan McGregor, is presently with Hull but required surgery on a back injury that had him sidelined at the end of last season.
Before the Hull’s interest emerged, head coach Paul Trollope admitted Cardiff had players who would interest other clubs.
”I think any manager or any coach wants to keep their best players,” he said. “But also I think any coach will tell you all players have their value whether they are top of the league or in League Two.
”Players do have their value. Of course you want to keep your best players, but we are also aware of their value.”
The signing of Amos and defender Joe Bennett, who has joined from Aston Villa, means Cardiff have made seven signings this summer.
Lex Immers, Kenneth Zohore, Emyr Huws, Jazz Richards and Frederic Gounongbe have also moved to the Welsh capital.
Meanwhile, midfielder Kagisho Dikgacoi’s contract with the club has been terminated by mutual agreement.