Ex-House Speaker Newt Gingrich called the Bush family “childish” while Mr Trump’s top aide said Ohio Governor John Kasish was “petulant”.
The nomination of Mr Trump has been a source of conflict in the party.
And there were attempts on the convention’s opening day to “unbind” delegates from supporting him.
A request by Republicans opposed to Trump to force a full vote on the convention rules was refused, causing brief uproar.
The long-simmering tensions between Trump and anti-Trump factions in the party broke into open warfare earlier on Monday.
Former House Speaker New Gingrich told ABC News “the Republican party has been awfully good to the Bushes and they’re showing remarkably little gratitude”.
The two former presidents, George HW Bush and George W Bush, have refused to endorse Mr Trump.
Mr Trump has criticised the younger Bush over the Iraq War and the 9/11 attacks, and constantly mocked Florida Governor Jeb Bush during his unsuccessful candidacy.
His top aide Paul Manafort launched a stinging attack on the governor of the state hosting the convention, the popular Mr Kasich, describing his decision to stay away as an “embarrassment”.
Former nominee Mitt Romney has also refused to attend, voicing concerns over Mr Trump’s tone and extreme stance on immigration.
Among the speakers due to appear on Monday evening are Mr Trump’s wife Melania and Iowa Senator Joni Ernst, on a theme of Make America Safe Again.
The convention begins amid high tensions, a day after a man killed three police officers in Baton Rouge, prompting Mr Trump to say the country was falling apart – a claim strongly disputed by President Barack Obama.
People are not allowed to take guns into Cleveland’s Quicken Loans Arena, which is hosting the four-day event, or within a secure zone outside.
But some people were photographed openly wearing guns nearby.
Cleveland police have asked Ohio’s governor to suspend open-carry gun rights during the convention but he said he did not have the power to.
Thousands of federal and state law enforcement officers have descended on the city over the past week in preparation for the convention, ramping up security protocols as delegates, attendees and demonstrators pour into Cleveland.
About 50,000 people are expected to travel to Cleveland during the four-day event, with protests and rallies expected to take place throughout the week.