The move rekindles a working relationship started when Bridges advised Ana Botín on her start-up Suala Capital in 2000 and which has included Quiller working for the bank.
The news comes after Quiller hired Sean Worth, who is understood to be stepping into Bridges' leadership role at the Huntsworth-owned agency.
Bridges was described as a "big loss" to the agency by one public affairs head.
Bridges told PRWeek: "I am delighted to be going to work at a great bank, in an interesting sector, for a dynamic and innovative CEO. I’m leaving the Quiller ship in good hands."
A friend of Chancellor George Osborne and a former leader writer for The Times, Bridges became Tory campaigns director in 2006 and also had a hand in its 2010 election campaign.
Known for his strong links to the Tory high guard, he has also served as chairman of the Conservative Research Department.
Botín, the daughter of Santander's global chief executive, became its UK CEO in 2010.
This year she was judged the third most powerful woman in the UK by Woman's Hour on BBC Radio 4, while in September she attacked British "grumbling" for slowing the economic recovery.
In the past nine months Santander has reduced mortgage lending from £156.6 billion to £149.9 billion, but increased its business lending from £19.6 billion to £21.5 billion.