Jessica Bridge of Ladbrokes said Friday that it seems an engagement announcement "is to be confirmed imminently."
The bookmaker has stopped taking bets on a 2018 royal wedding after Markle was seen shopping in London this week.
The British press has reported that Markle has already met in private with Harry's grandmother, Queen Elizabeth II.
The couple has been dating for more than a year and Harry has asked the press to grant them a certain amount of privacy.
Markle is believed to be in the process of moving to London.
Palace officials say they will not comment on the rumours.