The 20-year-old was given a late call up to the Three Lions squad over the weekend due to a number of withdrawals from Southgate’s regular options.
Cook was one of three players drafted in, alongside Liverpool’s Dominic Solanke and Norwich City loanee Angus Gunn.
Southgate has already given 11 players their debut in just over a year in charge of the national side and he has hinted that Cook could get his chance against Brazil at Wembley on Tuesday.
"Those (three new) guys have fitted in off the field no problem which helps them to fit in on the field - and they're good players, so in training they look fine,”he said.
"We have to see how the game progresses but I won't have any hesitation in putting them on the pitch.
"Angus has worked with the under-21s with us, he's playing in the league. The other two - captain of the Under-20 World Cup winners (Cook) and Golden Boot winner in the Under-20s World Cup (Solanke). They've got good pedigree, and we won't hesitate to put them on the pitch."