Rashid, who has been at Yorkshire since 2006, released a statement after the T20 World Cup and said he will be "happy to support" any further official investigations into Rafiq's claims.
Earlier this month, Rafiq had made allegations of racism against former Yorkshire captain Vaughan. He had claimed that Vaughan said "too many of you lot, we need to do something about it" to him and three other players before a match in 2009.
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"Racism is a cancer in all walks of life and unfortunately in professional sports too, and is something which of course has to be stamped out," ESPNcricinfo quoted Rashid as saying.
"I wanted to concentrate as much as possible on my cricket and to avoid distractions to the detriment of the team but I can confirm Azeem Rafiq's recollection of Michael Vaughan's comments to a group of us Asian players," he added.
The 2005 Ashes-winning captain Vaughan had admitted that he was the former player implicated in the investigation into Rafiq's allegations of institutional racism at Yorkshire but "completely and categorically" denied the charges.
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"I completely and categorically deny that I ever said those words. I have nothing to hide. The 'you lot' comment never happened," Vaughan, who was stood down from his BBC Radio 5 Live show, had written in a newspaper column.
"Anyone trying to recollect words said 10 years ago will be fallible but I am adamant those words were not used. If Rafiq believes something was said at the time to upset him then that is what he believes," he added.
The allegation against Vaughan was also earlier corroborated by former Pakistan bowler Rana Naved-ul-Hasan, who said: "systematic taunting" occurred at the club.
"I'm encouraged by the fact that a parliamentary committee seems to be trying to improve the situation, whether that's holding people accountable or getting changes made at an institutional level," the 33-year-old Rashid further said.
"These can only be positive developments. I will of course be more than happy to support any official efforts when the time is right. For now, though, these matters are of an intensely personal nature and I will not be commenting on them further. I ask you to respect my privacy and allow me to focus on my cricket. I want to thank the ECB, the fans, and especially my teammates for all of their support. We didn't get the result we wanted in this World Cup, but I hope that the unity of our dressing room and the leadership of our captain will propel us forward to achieve what we deserve in the future," he added.
Rafiq, who was born in Pakistan and moved to England aged 10, played the majority of his career at Yorkshire, between 2008 and 2018, and captained the club in 2012. In September 2020, the cricketer in an interview said that he encountered "institutional racism" at the club , which left him close to taking his own life.
Thereafter, Yorkshire launched a formal investigation into Rafiq's allegations and Yorkshire chairman Roger Hutton, who has now resigned said the club would be carrying out a "wider review" of their "policies and culture".
In August 2021, three days after Yorkshire received the findings of the independent report carried out by law firm Squire Patton Boggs, they admitted Rafiq was "the victim of inappropriate behaviour". The report summary said Rafiq was the "victim of racial harassment and bullying" at the club.
Yorkshire accepted the findings of the report but announced no disciplinary action would be taken against any employees. The county has been widely criticised for its response and the England and Wales Cricket Board (ECB) suspended their right to host international and major matches at Headingley.
The board also banned cricketer Gary Ballance from England selection for an indefinite period after he admitted using a racial slur against his ex-Yorkshire teammate Rafiq.