The 30-year-old goalkeeper has moved to the London Stadium from Manchester City on a temporary basis for the 2017-18 season, as he looks to secure regular first-team football after being deemed surplus to requirements by Citizens boss Pep Guardiola.
The experienced stopper insists his main priority during his time with the Hammers is to impress boss Slaven Bilic, as well as England coach Southgate.
"I'm trying to make a point to Slaven," said Hart. "I'm trying to make my point to Gareth Southgate. They're my coaches. They're the people whose opinions I care about.
"Gareth thinks West Ham is the right place for me to be. I want to play at the highest standard available to me. This definitely fits the criteria."
Hart has managed to retain his place as England's first-choice stopper for a number of years, but he knows he is under pressure to perform to the best of his abilities, with the likes of Jack Butland, Jordan Pickford, Fraser Forster and Tom Heaton all eyeing his place in the Three Lions starting line-up.
He added: "I've always said the standard of keepers in England is very high. It's a privileged position to play for the country. Every single time I've put on that number one jersey I have been very proud and very excited for what comes with it.
"You've got Jack Butland and Jordan Pickford now at Everton. Big Fras [Fraser Forster] has signed a new deal at Southampton. [Tom] Heaton has been nothing short of a miracle for Burnley with his performances."