Chelsea's Belgium international striker enjoyed an outstanding season-long loan at West Brom, where he netted 17 goals to finish sixth in the Premier League top scorers' chart.
The Blues are expected to send the 20-year-old out on loan again for the 2013/14 campaign, with Europa League debutants Swansea understood to be leading the chase.
However, Hammers supremo Sullivan believes he can convince Lukaku to remain in the capital and play at Upton Park alongside Carroll, who could be about to secure a permanent £15million switch from Liverpool.
Sullivan told the Daily Mirror: "Lukaku would be a loan again. We'd maybe buy him a year later if Chelsea decide they don't want him.
"If we sign Andy, it would be a record signing for the club. He and Romelu would be devastating. The best strike-force outside of the top six or seven.
"Lukaku has been amazing for West Brom this season and I'd hope that he would prefer to live in London."
Having surprised many pundits by finishing in the top half of the table in their first season back in the top-flight, Sullivan is now hoping the team can push on under Sam Allardyce.
"I am absolutely delighted with this season," added Sullivan. "To be clear of relegation with three or four games to go has been a lovely feeling.
"But we have another challenge ahead. Newcastle were fifth last year and yet hovered above the relegation zone this time. There can never be any room for complacency."