Senator John Cornyn, another veteran Republican, also said he thought any such effort "would be a bad mistake" soundly defeated in the 100-member chamber, currently controlled by his party.
Congressional negotiators have been trying for months to reach agreement on a new coronavirus aid bill, after Congress approved $3 trillion in relief earlier this year.
The $740 billion National Defense Authorization Act (NDAA) passed both houses of Congress with more than the two-thirds majority needed to override a presidential veto.
Organizers of Stop The Steal, linked to pro-Trump operative Roger Stone, and church groups urged supporters to participate in "Jericho Marches" and prayer rallies.
Pfizer's dry-ice cooled packages can hold as many as 4,875 doses, and the first leg of their journey will be from Kalamazoo to planes positioned nearby.
"We inadvertently took action to limit engagements on the labeled Tweet you referenced. This action has been reversed", a Twitter spokesman told Reuters in an emailed statement.