Speaking on Times Radio, Mr Bercow said the former prime minister had been too eager to “pay homage” to the outgoing president after he took office nearly four years ago.
Ms May was the first foreign leader to meet the US president after he came into power, with the pair holding bilateral discussions in Washington just days after his inauguration in January 2017.
She also invited Mr Trump to the UK for a state visit not long after his election victory, prompting an outcry across the country.
“She seemed almost breathless in her quest to get him to come to this country and speak here,” Mr Bercow, who stepped down as speaker in November 2019, said.
However, in the end, he said he believed Ms May realised that her efforts were “ultimately ineffective”.
Mr Trump visited the UK in June 2019, which sparked major protests, and she was replaced by Boris Johnson the following month.
Mr Trump was heavily critical of the former prime minister over her leadership on Brexit, championing her challenger instead, after their relationship soured when Ms May criticised his decision to retweet content from far-right political group Britain First.
“Her efforts on Brexit were fruitless and often maladroit, but [Mr Trump] then conducted a running commentary on those efforts, undermining her at every turn,” Mr Bercow said.
“So, I think she probably realised that currying favour with him was ultimately ineffective and maybe even counterproductive.”
Now, with his presidency almost at an end, Mr Trump faces widespread calls to either resign or be impeached for inciting the violence that unfolded at the US Capitol on Wednesday.
After pro-Trump rioters stormed the US Capitol Building in a deadly attack that left five people dead, the president has been banned from a number of social media platforms, including Twitter, from which he has been permanently suspended “due to the risk of further incitement of violence”.
An article of impeachment seeking to remove Mr Trump from office before his presidency ends is expected to be introduced on Monday.
While Ms May is facing criticism over her early efforts to build a relationship with Mr Trump, her successor, Mr Johnson, has also faced blame.
Mr Johnson condemned Wednesday’s chaos on Capitol Hill, but members of the Labour Party criticised him for failing to condemn Mr Trump’s involvement directly.
They also pointed to interviews in which Mr Johnson heaped praise upon the president, once suggesting he was as worthy of the Nobel Peace Prize as his predecessor, Barack Obama.