Sen. Ted Cruz raised almost $20 million for his presidential campaign in the fourth quarter — a 66% increase over the previous three-month period, according to campaign officials and a memo that shows how the Texas Republican’s fundraising pace accelerated as he has gained in the polls.

Jeff Roe, Mr. Cruz’s campaign manager, said in a memo to staff and supporters that the campaign’s year end fundraising total would come to more than $45 million, up from the $26 million raised by the end of September.

The $20 million haul in the final three months of 2015 is up sharply from the $12.2 million raised in the third quarter, a bonanza that came as polls showed Mr. Cruz rising to a top-tier candidate from a back-of-the-pack contender in the crowded GOP candidate field.