Media playback is not supported on this device
On Friday, the Jordanian Al-Qadi family bought a 92% stake in the promotion-chasing League Two team.
Rovers sit sixth in the table after coming from behind to beat Morecambe 2-1 on Saturday.
"If they want to sack me, they want to sack me, but as it stands at the minute the owner's fully behind me," Clarke, 38, told BBC Points West.
"I don't fear for my job - why fear for my job? As soon as you're a manager and you start fearing for your job, you're out the door.
"What I will say is that I have it a certain way. I'm the manager, I pick the team, I pick the players that come into the club and the new owners are embracing that."
The Pirates are five points outside the League Two automatic promotion places, having secured their return to the Football League last May.
But Clarke, who replaced John Ward as Rovers manager in March 2014, said he did not find out about the takeover until the morning the deal was announced.
"We never knew about a takeover," he said. "You hear rumours and all the bits and pieces, but we found out about this on Friday morning and I'm supposed to be preparing my team.
"It's great what the owners have been saying and how the club's going to move forward, it's great for the supporters. But on the back of that, I've got to manage that.
"I've got staff who are thinking: 'Well, what's going to happen with my job?' You've always got people that are concerned about their jobs."
Bristol Rovers boss Darrell Clarke says he is not worried about losing his job following the club's recent takeover.