The 19-year-old winger was born in Liverpool and has represented England from Under-15 through to Under-20 level, but remains eligible to play for Ireland through his father, Paddy.
Lacey has recently attracted attention after making an impact for Manchester United. He provided the assist for Matheus Cunha's winning goal in Tuesday's 3-2 Champions League comeback victory against Roma, becoming the youngest United player to assist a Champions League-winning goal.
Hallgrimsson was asked whether he had been in contact with the teenager ahead of Ireland's forthcoming Nations League fixtures against Austria and Kosovo. The manager suggested that any approach would be handled by other members of the Football Association of Ireland.
"That would be for other guys in the association," he said. "But it was interesting looking at his Irish connection."
Lacey's recent progress at Old Trafford has strengthened his profile. He made his first-team debut in a Carabao Cup defeat against Brighton in September, scoring after seven minutes, and has since registered one goal and one assist in ten appearances across competitions.
He came off the bench for the final 16 minutes against Roma and quickly made his mark, setting up Cunha's decisive strike as United recovered from two goals down.
Ireland have a potential connection to the player through their coaching staff, with former Manchester United defender John O'Shea serving as Hallgrimsson's assistant. The manager has also confirmed that former Everton full-back Seamus Coleman will work with the national team in a coaching capacity.
Lacey has yet to make a senior international appearance, meaning his future remains open despite his extensive involvement in England's youth teams. Ireland's interest could therefore develop into a contest for his international allegiance as his club career gathers momentum.