Bellingham, who was appointed Africa minister in May when Prime Minister David Cameron’s coalition government came to power, said Britain will use its influence in southern Africa to push for free and fair elections in Zimbabwe.
"Our position is very simple. We want to see credible elections and we’re going to use our influence to support those people who are working in that direction," Bellingham told journalists in Johannesburg at the start of his first official visit to regional power-broker South Africa.
Bellingham was speaking the day after Mugabe asked leaders at a summit of African and European Union states why former British prime minister Tony Blair and former US president George W. Bush had not been indicted for war crimes like Sudanese President Omar al-Bashir.
Mugabe said the two former leaders were responsible for hundreds of thousands of Iraqi deaths.
"He does come up with comments from time to time, doesn’t he?" Bellingham said. "I think one just has to take him with a pinch of salt in that particular case."
Mugabe has long had a tense relationship with former colonial ruler Britain.
Blair said last year that Zimbabwe should "get rid of" the 86-year-old leader, saying Mugabe "has destroyed his country".
Zimbabwe is currently ruled by a power-sharing government formed between Mugabe and long-time rival Morgan Tsvangirai in the wake of bloody and inconclusive elections in 2008.
Under the deal, the country is supposed to adopt a new constitution that will steer it to fresh elections.
Mugabe’s ZANU-PF party has said new polls will be held around June next year, even if the stalled constitution-making process is not complete.
But Tsvangirai’s MDC party has called for key reforms to be implemented first to ensure a free and fair election.