In 2013 in response to the Republicans rejecting President Obama's nominees, Senate Democrats reduced the vote to halt a filibuster to confirmation of certain government positions to 50 votes. Now, in response to the threat of democrats blocking a President Trump's SCOTUS nominee, Republicans are floating the idea of reducing the cloture to 50 votes for SCOTUS positions as well.
Is it possible for Congress to restore cloture rules back to the way they were, where it requires 60 votes to approve nominees? Please note, I'm not asking if the cloture will be restored, but whether through congressional procedures it can be restored.
Of course. Anything Congress can change Congress can change.
2: Each House may determine the Rules of its Proceedings, punish its Members for disorderly Behaviour, and, with the Concurrence of two thirds, expel a Member.
It seems pretty clear this is a rule of its proceedings
The thing they can't change is that it has been changed before, and that means politicians will consider changing as an option.
Possibly if enough people from their own party made enough of a stink about it they might think it unwise, but they will always have the option to change and change back their rules.
The problem with changing the cloture rules via a rule is that the rule can be reversed again. That was the real evil of the "nuclear option." Now, any Senate rule can be changed by a simple majority vote (actually, it always could but no one broke the rules that way). So they change cloture back to sixty votes. Someone filibusters. They change cloture to a simply majority, which they can do with a simple majority vote. This effectively makes the cloture always require a simple majority to invoke.
There's some minimal value to having the rule be at sixty anyway. So long as the custom holds, the vote can remain at sixty. But the ability to change it at any time means that it is only a minimal value.
If they really wanted to change things to require sixty votes again, they'd need a constitutional amendment. Whatever they do could be changed the same way it was instituted. At least with an amendment, the process takes years. They couldn't trigger it just to get a single cabinet official or judge confirmed.