The phrase "persistent rumours" means that there are rumors or stories that keep going on for a long time and refuse to go away or be proven false. Full definition
The British film industry is in rude health, according to the film council, persistent rumours of its imminent demise notwithstanding. (theguardian.com)
But while he was away, there were persistent rumours online that he was seriously ill or even dead, despite photographs showing him meeting senior Nigerian politicians. (nigeriapoliticsonline.com)
Ewan was the subject of persistent rumours about on - set flirtations, but always denied rumours of a fling with Nicole Kidman, his co-star in Moulin Rouge, and Renee Zellweger, in Down With Love and Miss Potter. (dailymail.co.uk)