The Saints have endured a difficult start to life back in the Premier League under Martin following promotion from the Championship via the playoffs in May.
Collecting just the single point from their opening seven contests, Southampton are languishing down in 19th place in the top-flight standings.
Prior to the October international break, the Saints have suffered back-to-back 3-1 defeats at the hands of Bournemouth and title challengers Arsenal.
Cameron Archer fired Southampton ahead after 55 minutes at the Emirates Stadium on Saturday, however the Gunners staged a comeback courtesy of efforts from Kai Havertz, Gabriel Martinelli and Bukayo Saka.
Southampton to decide Martin's future soon?
According to Football Insider, Southampton are reviewing the position of under-pressure coach Martin over the course of the October international break.
The report claims that the Saints' hierarchy are preparing for a meeting, in which the future of their 38-year-old manager will be discussed.
Despite a sobering start to the campaign, it is understood that Martin has credit in the bank with his employers due to the way that he led the club back to the Premier League last season.
At this stage, the Scotsman will be supposedly afforded until the next international break in November to turn around the fortunes of the South-coast club.
There is said to be a lack of available managers currently, incentivising Southampton's owners to stick with Martin for at least another few weeks.
Southampton's upcoming fixtures
If the reports are accurate, Martin could have only four Premier League matches to save his job at St Mary's, as well as an EFL Cup clash with Championship side Stoke City.
Southampton will be targeting their maiden victory of the top-flight campaign when newly-promoted Leicester City visit the South coast immediately after the international break.
Things could turn sour quickly for Martin if his team fail to record a positive result against the Foxes, with a trip to Manchester City scheduled for October 26.
The Saints will then round off a quartet of Premier League battles with the hosting of Sean Dyche's Everton and a match at Wolverhampton Wanderers.