I didn’t know that Presidents had the power to prosecute anyone.
Article II of the US Constitution says, "the executive power shall be vested in a President of the United States.”
The Attorney General authority is delegated from the President so yes, a President can prosecute someone.
This is a really interesting question.
The President is the Chief Executive Officer of the Department of Justice, which includes the FBI.
The Attorney General probably has the final legal authority to prosecute, and the President nominates - but does not confirm - the AG.
On the other side, the President probably does have the final authority to STOP a prosecution, since the President can fire any appointed DOJ executive who tries to enforce a prosecution.
I will speculate that the President, based on criminal evidence, can order the FBI to arrest anyone, but, if all the appointed FBI executives resign, arrests, and prosecutions, might be impossible, unless the Senate confirms DOJ executives who are willing to obey the President.