The 25-year-old -- who shone at last year's World Cup -- is believed to have reacted when revellers in a bar directed derogatory remarks at his fiancee Megan, who recently had their first child Arlo.
A video has spread on social media in which onlookers can be heard shouting "Jordan" as a fight breaks out before the man, believed to be Pickford, is dragged away.
Pickford had played in Everton's 2-0 away win over West Ham on Saturday before spending a day off in his home city.
"The club has been made aware of an alleged incident involving one of our players and we are looking into the matter," the Premier League club said.
Pickford came through the system at Sunderland but joined Everton for an initial £25 million ($32.6 million) when they were relegated from the Premier League in June 2017.
He made his England debut five months later and was Gareth Southgate's first choice at the World Cup and was instrumental in their surprise run to the semi-finals.
He saved a spot-kick in the penalty shootout victory over Colombia in their last 16 clash and was man of the match in England's 2-0 win over Sweden in the quarter-finals.