Cookson has been president of the UCI - the world governing body for competitive cycling - since 2013, when he vowed to restore credibility to the sport.
For years cycling has been plagued with doping revelations, including disgraced former cyclist Lance Armstrong admitting to using performance-enhancing drugs to win each of his seven Tour de France titles during an interview in 2013.
In 2015, a report by the Cycling Independent Reform Commission found evidence that the UCI had colluded with Armstrong between 1999 and 2009 to bury the allegations.
If re-elected, Cookson said he would build on the restored credibility he has overseen since taking office.
Calacus' remit includes handling PR for Cookson's election campaign and helping set out his vision for the next four years, agency MD David Alexander said.
Alexander said: "Cycling has gone through some challenging times over the years and the fact they have now been handed four more medal events at the Olympic Games in Tokyo in 2020 is a testament to the progress the UCI has made under Brian Cookson's leadership."
Cookson is running against one other candidate, David Lappartient, who is a UCI vice president. The election will take place in Bergen, Norway, in late September.
Calacus, which handles PR for a range of sports administrators, rights holders and sponsors, counts the Red Bull Air Race and the International Boxing Association among its recent clients.