The number of patients going to A&E at Pinderfields Hospital in Wakefield and Dewsbury Hospital has risen to unprecedented levels, a public meeting was told on Thursday.
It follows the Yorkshire Ambulance Service declaring a major incident last week and pleading with the public to only call in the event of a "serious or life-threatening emergency".
The Mid Yorkshire Hospitals NHS Trust, which runs the Pinderfields and Dewsbury sites, has repeatedly warned about A&E numbers creeping up for several months.
But attendances in recent weeks have broken new ground and hospitals elsewhere in the region are struggling to cope too.
Speaking at a trust board meeting, chief operating officer Trudie Davies said: "The key area of pressure is in emergency care.
"The attendance rate has been significant. We've had the highest month's attendance in the history of Mid Yorkshire across May and June.
"The pressure on the organisation is really very significant.
"That pressure is being felt in all of our neighbouring trusts."
Ms Davies said that the high numbers of inpatients meant it was "very difficult" to maintain social distancing in A&E.
She added: "Lots of patients are attending with (less serious) needs.
"They're having difficulty in accessing primary care appointments.
"Our primary care colleagues are working extremely hard. This isn't a reflection on the work they're doing, but the demand in the system is high."
Healthcare professionals believe the current issues stem from lockdown, where hospitals were inundated with Covid cases and patients with less urgent needs were advised to stay away by the government.
The board meeting was also told that non-emergency surgery has also been affected at Pinderfields Hospital, because of a shortage of theatre staff.
Local Democracy Reporting Service