Plans have been drawn up three times in the last decade with a view to redeveloping the Sir Bobby Charlton Stand, which would take the stadium capacity to 88,000.
However, Arnold claims that an expansion would prove "a very complex engineering challenge to deliver" and is also concerned about what ground could be used in the interim.
"We haven't left the easiest stand until last," Arnold told United We Stand, quoted in the South China Morning Post. "The South Stand - the Sir Bobby Charlton Stand - is the last unexpanded stand.
"It's sandwiched in a fairly narrow strip of land between the pitch and the main Manchester to Liverpool railway. There's not much space on either side of the stand, meaning that it's a very complex engineering challenge to deliver.
"It would be a multi-season challenge and it isn't certain that there's a way of doing it which doesn't render us homeless. We have the biggest fanbase in the country and we're not in London, where we could use Wembley.
"So that's not a light challenge for us and we don't want to give up the home advantage that we have here. Moving elsewhere would be difficult and the logistical issues associated with expanding the South Stand are not negligible. And that's before you start getting into how big the expanded stadium should be."
From the start of next season, Old Trafford's capacity will fall from just under 75,000 to 73,300 because of work required to accommodate higher numbers of disabled supporters.