Lancaster was chosen as caretaker coach following the resignation of Martin Johnson after the Rugby World Cup last year.
"Stuart has made a fantastic impression and I'm sure if Stuart chooses to apply he will be given every consideration for that job," Brown said.
"Stuart has to make the decision about whether he wants to be considered for the role.
"The door is not shut, it's open. All of the good things you have seen Stuart has done have made a massive impact and that is the kind of thing you want."
Lancaster's new-look England take on Scotland in the Six Nations on Saturday, February 4.