The midfielder is available on a free having left Zenit St Petersburg this summer.
Despite the copious options in midfield, Steven Gerrard tried to bring the seven-time Serie A winner to Ibrox, according to Italian journalist Nicolo Schira.
However, the 33-year-old rejected the chance to join Rangers.
His arrival would have brought the number of signings this summer at Ibrox to 11 and he would have been the eighth central midfield option.
Marchisio played nearly 400 times for Juventus, winning a glut of trophies, before making the move to Russia where he helped Zenit win the league last season.
The 55-cap Italian international may have raised some eyebrows among the Rangers support, both for his versatile style of play but also because there is a picture of him holding up a t-shirt supporting FARE, the organisation which, along with Uefa, raised the issue of sectarian singing during European matches.