The Senate because the bar for opening an investigation in the Senate is much lower than the bar for the USDOJ opening an investigation.
The Senate only requires a handful of Senators to desire an investigation.
The GOP controlled Senate could have an investigation going at the whim of fewer than a dozen GOP Senators.
The Senate can investigate the actions of civilians.
Congress can investigate most anything and most anyone at most any time. There are few practical limits on their powers of investigation.
No man's life, liberty or property are safe while the Legislature is in session
The DOJ requires reasonable evidence etc.
A much higher bar.
fwiw, Trump has already done more than he can to investigate the matter— one of the reasons why he's in trouble atm.