"President Trump is once again proving he is the MVP of trolling his competitors," he said.
In the interview, Trump said the Delaware Democrat, "seemed to have gotten along" with the former president while the two were in the White House.
“How he doesn’t get President Obama to endorse him, there has to be some reason why he’s not endorsing him,” he said.
"And then he goes and lies and says I asked the president not to endorse me - give me a break - because he's embarrassed by the fact that Obama is not endorsing him.
"So he goes out and says I asked President Obama not to endorse me."
Shillue claimed Trump's comments showed a keen sense of strategy.
"This is politically brilliant because he's going at the big strength of Joe Biden - which is his association with Obama," he said.
"He's creating mystery. The big mystery is what it's all about."
Trump and Biden repeatedly have traded blows since the former vice president entered the 2020 race, with the president frequently dismissing his potential opponent as “sleepy”.
Trump even said he would prefer to run against Biden rather than face another campaign against former Secretary of State Hillary Clinton.
Fox News' Liam Quinn contributed to this report.