The two then talked about how stressful was to watch the team at the minute and they also joked about the referee, Michael Oliver, who disallowed three West Ham goals.
Hart was outstanding in the Hammers' previous game against Chelsea but, against Stoke in the 79th minute, he had failed to hold a shot from Xherdan Shaqiri and Peter Crouch was on hand to score from close-range.
- Andy Carroll has scored three goals in his last three Premier League appearances, spanning January (two games, two goals) and April (one game, one goal).
Stoke was close to a huge win in its fight to avoid the drop when Carroll, who had been on the field as a substitute for just four minutes, met a left-wing cross with a crisp volley into the bottom corner.
Their hosts for this fixture find themselves in a much brighter situation, however, things are running far from smoothly for the Hammers.
West Ham were searching for back-to-back Premier League victories at the London Stadium for only the second time and threatened within the opening moments as Stoke goalkeeper Jack Butland repelled Arthur Masuaku's low drive with his feet.More news: WPP chief resigns after allegations of 'personal misconduct'
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The goal was Crouch's 44th for Stoke, making the big man the club's highest all-time scorer in the Premier League - moving him one ahead of Jonathan Walters on 43.
Southgate was at the London Stadium on Monday night to run the rule over Hart, who was named in West Ham's starting line-up against Stoke, and Butland.
West Ham United are set to show Jordan Hugill the exit door this summer after a short stint with the club and fans are utterly bewildered by the news.
While going into the game both sides would have been eyeing the win, Stoke needed the points more than West Ham in their quest for safety. Over their last 10 games, Stoke have picked up seven points. "We've just had a win and two draws and at this time of the season that's not too bad".
West Ham still have five league games remaining this season, and Carroll will be hoping to finish the season in fine style.
Hart will no doubt have been mortified by his costly error, but Carroll is confident his Hammers colleague has what it takes to bounce back.