WASHINGTON—Plans for a Democratic presidential debate in January have been complicated by the expected Senate impeachment trial of President Trump, which could tie up several 2020 hopefuls during the final stretch before the Iowa caucuses.

The Democratic National Committee has struggled to settle on a date for next month’s debate, according to people familiar with the planning. It isn’t clear when a potential trial would start or how long it would last; the Senate trial of former President Bill Clinton was conducted over five...