UK ministers have written to Mark Drakeford to offer "mutual aid" in England if Welsh hospitals become unable to treat non-Covid patients.
It could see coronavirus patients being moved to hospitals in England to ease the pressure on hospitals in Wales.
It comes as numbers of patients in Welsh hospitals with Covid-19 hit record levels.
The first minister said Wales has supplied 11 million items of PPE to the English NHS during the pandemic.
Two Welsh health boards have suspended some non-urgent care in response to increased coronavirus cases.
Wales is already in breach of some key indicators used to determine future lockdowns, including current coronavirus case rates and the rate of positive tests.
Concerns over hospital capacity is also a key indicators for Wales's new alert level four - which would introduce restrictions equivalent to a lockdown.
England's Health Secretary Matt Hancock and Welsh Secretary Simon Hart have written to the first minister to offer support for Welsh health boards and said the military would assist with logistical support.
Covid 'does not respect boundaries'
They wrote: "We have seen that Covid-19 does not respect political or administrative boundaries. Instead it follows the human geographies that people follow, which we know is especially important for much of the Wales-England border.
"We are aware that health boards across Wales are considering postponing non-Covid treatments and appointments (including those classed as urgent) to deal with the immediate threat of the pandemic.
"We stand ready to support Welsh hospitals through mutual aid of patients across the border where this is needed due to undue strain on healthcare provision."
The letter also offers further use of the UK government's mass testing programme - in addition to current mass-testing programmes in Merthyr Tydfil and Cynon Valley.
They added the military "stand ready to assist with the Welsh Government with further logistical and other support where needed too".
How many patients are in hospitals in Wales?
The latest NHS Wales figures show 2,124 patients with Covid in hospital on Monday - this is 17% more than the week before and the highest number yet recorded.
It has risen over the week in all areas apart from Betsi Cadwaladr, while Aneurin Bevan has the most patients, 584.
Cwm Taf Morgannwg health board has seen a 30% rise in Covid patients.
Numbers include more than 600 patients in Welsh hospital beds who are still recovering from the effects of the virus.
Covid-19 patients make up nearly 27% of all patients in hospital, a proportion which has been slowly rising in the last week.
There were 91 patients in critical care or on invasive ventilation, which is the highest number since the first wave of pandemic in May - with 48 more patients in total than the usual capacity,
Hospital admissions remain fairly steady with an average of 90 a day for the last week.
Lee Waters, the deputy minister for economy and transport, said on social media the Welsh Government had "provided repeated mutual aid to all other parts of the UK to stop them running out of PPE".
He accused Simon Hart of "cheap posturing in a public health emergency... shame on you".
First Minister Mark Drakeford said he would reply to the letter "once again endorsing the principle of mutual aid which has been there through coronavirus".
"Wales has supplied 11 million items of PPE to the English NHS during the virus as part of our mutual aid," he said.
"The letter that I have received today restates that and I'm very happy, once again, to endorse it."