There is no constitutional provision which grants the President such power. I am not aware of any provision of federal law which grants such power, nor of any case where a real president has exercised such a power. Of course, the President is a citizen, and any citizen may make a Citizen's arrest Particularly for a felony. (See also this FindLaw page on the subject.) And of course, as the head of the executive department, the President could order someone who clearly has powers of arrest to make an arrest, although such an order would not be valid in the absence of legal cause to make an arrest.
A comment called atention to 10 USC §252 which provides:
Whenever the President considers that unlawful obstructions, combinations, or assemblages, or rebellion against the authority of the United States, make it impracticable to enforce the laws of the United States in any State by the ordinary course of judicial proceedings, he may call into Federal service such of the militia of any State, and use such of the armed forces, as he considers necessary to enforce those laws or to suppress the rebellion.
The history for this section dates from 1861. A note indicates that a prior version was the basis of Executive order 10730, sending federal forces into Little Rock, AK in 1957 to enforce school integration there.
However, this section does not explicitly increase the arrest powers of anyone, and does not grant the President personally any power of arrest.