Mr Cummings is accused of breaking coronavirus lockdown rules by making a 250-mile journey to Durham with his family, when his wife had coronavirus symptoms.
Downing Street has defended Mr Cummings' actions, insisting he "acted in line with guidelines" and that his journey was "essential" because it related to the welfare of his child.
Boris Johnson's chief adviser, who was not ill at that point, has also said he did the "right" thing.
In a statement on Saturday, West Midlands Police and Crime Commissioner David Jamieson said "people will now say there is one rule for them and one rule for the Prime Minister’s political advisers".
PCC responds to reports of lockdown breach by Government Special Advisor:
“People will now say there is one rule for them and one rule for the Prime Minister’s political advisors.https://t.co/ihopLANb2n pic.twitter.com/DB9pfQRx5a— WestMidsPCC (@WestMidsPCC)May 23, 2020
The former Labour MP has called for Mr Johnson to sack Mr Cummings to "restore public confidence".
“The police’s job of enforcing the lockdown has been made much harder after both the actions of Dominic Cummings travelling over 260 miles and the flexibility with which the Government now seem to interpret the guidance," he said.
“People will now say there is one rule for them and one rule for the Prime Minister’s political advisers.
“Across the country, millions of people are making sacrifices, including scores of police officers and staff who have separated from their families to protect their health and the health of the nation.
“It seems beyond belief that to justify the breach of guidance they appear to now be questioning the honesty and credibility of Durham Police.
“The Prime Minster, Boris Johnson, should sack Dominic Cummings forthwith, to restore public confidence and some credibility to his handling of this dreadful Covid-19 crisis.”
A snap poll by YouGov found that 68 per cent of Brits think Mr Cummings broke coronavirus rules by taking his family to Durham, while 52 per cent think he should resign.
The survey of 3,707 adults, carried out on Saturday, showed just 28 per cent think he should stay on, with 20 per cent on the fence.
Meanwhile, the poll revealed Conservative voters were split over whether he should retain his position, with 41 per cent of those who responded saying he should quit while 43 per cent want him to remain as senior adviser to Boris Johnson.
Chris Curtis, political research manager at YouGov, said: “These are clearly troubling numbers for the Government and Mr Cummings.
"The public already thought that the Government was too hasty in lifting parts of lockdown and it’s likely they will be even less impressed if key public figures are perceived to have broken rules they think are not strict enough.
“Of course, the Government will be hoping everyone quickly moves on from a story about a relatively unknown adviser, but it’s going to be tough when the public thinks he was wrong and that he should go.”
Speaking to reporters outside his house on Saturday evening, Mr Cummings said he will not be considering his position. "Obviously not," he told journalists.
"You're not going to consider resigning? The public are probably very angry," one reporter asked.
He replied: "You guys are probably as right about that as you were about Brexit - remember how right you all were about that?"
It comes as Grant Shapps suggested it is up to each individual to decide how best to follow lockdown measures, as he joined a chorus of support for Mr Cummings over his trip during lockdown.
Facing a deluge of questions at the Downing Street daily briefing about whether Mr Johnson's chief aide had flouted the Government's own guidelines, the Transport Secretary insisted people should adhere to the rules "to the best of your ability" but that there might be times when "not all these measures will be possible".
The minister said he didn’t know the “personal circumstances” of the Cummings family that prevented them seeking help closer to home, as opposed to travelling to Durham.
He said: “As we all do in moments of crisis, we always seek to have our family, those who can assist us around us and I think that’s all that has happened in this case."