Shapps officially launched his leadership campaign over the weekend and pledged to deliver on the goals of outgoing leader Boris Johnson, but end a period of "often-distracted" government.
Speaking to Sky News, Eustice said all of the potential candidates have "got their strengths," but Shapps "is the best man" to take up the role at this particular point.
"He has got cabinet experience over several different leaders, he has shown very good judgement in my view and he is very good at communicating," he added.
"He did a stint as party chairman so he’s a good campaigner, and we’re going to need someone who can campaign as we go into the next election. He has got all of the strengths and experience we need at this particular time so that’s why I’m backing him."
