The 22-year-old was linked with a move to France this summer and more recently to Portuguese giants Benfica, but Samaras is believed to favour staying in the Premiership.
Boro have money to spend after selling Yakubu to Everton and their assistant manager Malcolm Crosby confirmed they are interested, although he feels it could be too late for the deal to go through.
"We've shown an interest in him and we'll continue to do that," he said.
"At the minute we are not really close to signing anyone. There are one or two players we're interested in, but it's looking like we might run out of time."
Crosby also claimed Boro have been offered Adriano on loan but said they would not be pursuing the City target.
Meanwhile, City defender Danny Mills is set to complete a loan move to his former club Charlton.
The former England international is surplus to requirements at City and the Blues are eager to get him off the wage bill.