Trump's support for Moore has left Senate Republicans dumbfounded after they tried to force the nominee out of the race after the Washington Post reported this month that when Moore was in his 30s, working as a local prosecutor, he is alleged to have pursued sexual relationships with teenagers - including a 14-year-old who said she had a sexual encounter with him.
After keeping quiet about Moore's scandals for weeks, Trump told reporters last week that his top priority was making sure his party kept its Senate seat.
Despite the multiple allegations against Moore - he reportedly was banned from an Alabama mall for repeatedly bothering teen girls - Trump has defended him.
Jones said he would be an independent vote in the Senate.
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Spokesmen for Trump and Pence declined to comment and referred questions to the office of Defense Secretary Jim Mattis. The communications agency employs 1,200 staffers drawn from all branches of the military.
On CNN, Sen. Lindsey Graham (R-SC) was asked about Trump's support for Moore, saying, "That's a political decision by the president".
"The moral of the story is: Don't nominate somebody like Roy Moore who could actually lose a seat that any other Republican could win", Graham said. Because he "is a Schumer/Pelosi puppet who is WEAK on Crime, WEAK on the Border, Bad for our Military and our great Vets, Bad for our 2nd Amendment, AND WANTS TO RAISES TAXES TO THE SKY". He totally denies it.
"The president can speak for himself, as far as I'm concerned, the president can use his influence and do what he can to get Moore to step aside", Mr Thune said.
Moore has lost almost all support within the party, but Trump appears to be sticking by his man, despite campaigning for his opponent, incumbent Senator "Big Luther" Strange, in the runoff Republican primary. He shot way up in the polls but it wasn't enough.
Thune said, however, if Moore wins his race "there's going to immediately be an ethics investigation which is going to be a cloud that he'll be operating in". "Liberal Jones would be BAD!" Luther Strange on Twitter, then deleted some of those tweets after Strange lost the runoff in September.
Trump's tweets on Sunday aren't his first swipe at Jones. I've looked at his record. It's bad on military. Trump was annoyed by his daughter's comments and said to Oval Office aides, "Can you believe this?" according to The New York Times.