Ally McCoist says Rangers have yet to decide whether to improve their offer for St Johnstone midfielder Murray Davidson.
The Perth club rejected an offer for the 23-year-old from their Scottish Premier League rivals on Thursday.
"I don't know to be honest," the boss said when asked about another bid.
"I haven't had a chance to sit down and speak to the owner at any great length, so it would be wrong to give a comment on that at the moment."
McCoist has the ability to register one more signing ahead of Tuesday's Champions League qualifier against Malmo but appears to have made a central defender his priority.
However, while he has already recruited Juan Manuel Ortiz from Almeria and Alejandro Bedoya from Orebro to strengthen his midfield, the latter will not arrive until January.
Former Scotland Under-21 international Davidson moved to McDiarmid Park from Livingston in May 2009 in a two-and-a-half year deal and has become a key player for the Saints.
The 23-year-old has twice been called up to the full Scotland squad.
A statement on the Perth club's website stated: "The offer from the Ibrox club fell short of the valuation placed on the player by chairman Geoff Brown."