Illinois is one of dozens of states where governors oppose accepting Syrian refugees.
The Iowa Republican wants Defense Secretary Ashton Carter to look into the allegations.
Sen. Chuck Grassley's office said the Iowa Republican's source is "well-known" to Democratic staff.
Democrats are pushing to block those on the terror watch list from buying guns.
Lawmakers have focused on the visa waiver program in the wake of the Paris attacks.
The majority leader started the fast-track process for a House-passed bill.
Paul's legislation would require the District of Columbia to grant concealed carry permits.
Sen. Jack Reed (D-R.I.) blocked McCain's request with "great reluctance."
The presidential candidate's push thwarted a housing and transportation bill.
Keith Laing 11/19/15 01:14 PM EST
The measure extends transportation funding until Dec. 4.