It is believed Sterling was unhappy with Gomez following an altercation with the Liverpool defender during City's Premier League defeat at Anfield last weekend. "I find myself just watching him and thinking "wow".
Kane took his tally to 31 goals for England with a clinical hat-trick against Montenegro, taking him above Alan Shearer, Nat Lofthouse and Preston North End legend Sir Tom Finney on the all-time list.
"I then spoke to Joe privately about how he was feeling".
"I've taken full responsibility and accepted the effect".
Despite Sterling accepting responsibility for the initial confrontation, Gomez still felt the brunt of the anger from supporters.
"I think now is a good moment for him just to go home for a couple of days, clear his head".
"I don't know what more I could've said", he said.
It was also reported that Gomez was the first person in and out of the Three Lions dressing room at full-time. No England player should ever be booed when they are wearing the shirt. "We have a young team who really play with freedom and expression, thinking about what's possible on the field rather than worrying about losing the game". I don't get it.
"Him and Raheem are very close and the whole team are very close so we don't want to see anybody treated in that way". I think we all handled it as well as we could have. "It's been a hard week but one that will make him stronger and one that he's dealt with really well".
Asked whether he will speak to Gomez about the issue, Southgate said: "Absolutely". We get Raheem back on the pitch tomorrow. "I've already started that, and yes, he's got to feel the support of everybody, which he has, completely".