Bower will join the Foreign Office in a senior comms role imminently. She had worked in Number 10 in various roles since May 2011.
It has been rumoured and reported in some media that Bower will be succeeded by Daily Mail journalist James Slack, but this has not been confirmed.
Last year, the Civil Service Jobs website advertised for a new spokesperson for the PM on a salary of £87,000, with a closing date of 13 January. Interviews for the post were due to conclude last week.
Swift, meanwhile, is moving to the the Department for Exiting the European Union, although it is thought he will not move until a successor has been found for his Downing Street role.
Former Daily Express journalist Swift joined Number 10 under Prime Minister David Cameron in April 2016.
He will oversee a DEXEU comms team of around 23, most of whom are media relations staff, and a ministerial correspondence and FOI team of around 21.
Rae Stewart, who had been interim head of comms since the department was set up in the summer, is understood to be leaving the department once a suitable handover to Swift has been arranged.
The Number 10 press office did not respond to PRWeek's request for comment before this article was published.