The Giants are expected to start GM interviews next week, according to Judy Battista of NFL.com (on Twitter). However, they are unlikely to name one before end of the season for fear of missing out on candidates currently with teams. 
After Jerry Reese was let go, the Giants did not rule out the possibility of signing his replacement before the end of the calendar year. However, that was always a longshot. By waiting until the offseason, the Giants will give themselves an opportunity to gauge a wider range of executives and give themselves additional time to make the right hire.
The Giants are expected to start with former Panthers GM/Giants exec Dave Gettleman and their own interim GM Kevin Abrams, according to Ralph Vacchiano of SNY (on Twitter). After that, he hears that the Giants will likely meet with two or three other candidates in early January.
Gettleman is said to be the frontrunner for the job, though Abrams will also be given a legitimate chance to be the team’s GM going forward. It sounds like former Chiefs GM John Dorsey also would have been in the mix for the job if the Browns did not pounce on him first.