Separately, Fredrik Gustavsson - previously easyJet head of strategy and M&A - has joined Teneo as an MD in the London office.
It's understood Claydon has taken a senior role in London, although his job title has not been revealed.
Claydon joined Brunswick in 2001, initially as a partner. He oversaw the London office from 2010 until 2014, when he was replaced by Simon Sporborg in a shakeup of the management team.
For the past five years, his role has been integrating the group’s client offer across Europe. Before joining Brunswick he was head of communications at British Airways.
PRWeek asked Teneo about Claydon's appointment but had not received a response at the time of publication.
In a separate statement, Teneo said new hire Gustavsson will have a particular focus on the European aviation market, working alongside Matt Lovering, senior MD and global transport and aviation consulting lead.
Gustavsson previously spent seven years at easyJet and also held senior positions at Credit Suisse and McKinsey & Co.
Their appointments follow that of Sally Osman, former communications director at the Royal Household, who joined Teneo's senior advisory group this summer.
Teneo is a few months into its new ownership structure. In June, private equity company CVC Capital Partners announced that it had taken a majority stake in agency, replacing BC Partners after four-and-a-half years.
Recent hires at Brunswick include Duncan McCourt, former chief of staff at the Treasury and special adviser to ex-Chancellor Phillip Hammond, who joined as a partner last month. In February, the agency announced six new partners in its London office amid a number of promotions globally.