The Spanish international is part of a 28-man United squad currently based in Perth, Australia - the location for the Reds' first pre-season fixture of 2019/20, which takes place this Saturday (13 July) against Perth Glory (kick-off 12:00 BST).
De Gea has just one year remaining on his existing deal, but Solskjaer is confident the 28-year-old's future lies at Old Trafford.
“David's had a summer now where he's been negotiating and thinking about his contract, so we hope that we're going to get it sorted soon.”
The Madrileno joined United in 2011 from boyhood club Atletico Madrid, and has since made 362 appearances and claimed the Sir Matt Busby Player of the Year award on a record four occasions.