Ashton also ran in the 2012 campaign to succeed the late Jack Layton.
The candidates are set to debate each other for the first time in Ottawa on Sunday, but the race does not end until October.
WATCH: Mulcair says he has ‘options on the table’ but will remain NDP leader until the fall
Ontario deputy NDP leader Jagmeet Singh is still eyeing a possible leadership bid, but his office says he is not planning to make his plans known this week.
Some of the central challenges for the party include the need to confront a $5.8 million debt and to boost the number of card-carrying supporters, which was down to 60,000 as of December.