NHS Lanarkshire said that increased Covid-19 cases, staff shortages due to annual leave or staff having to self-isolate as a result of contacts outside of work, were causing the pressure, along with trying to maintain and recover services - including planned operations.
On Monday 643 patients attended accident and emergency departments across the three hospitals.
NHS Lanarkshire director of acute services, Judith Park, said: "The sustained pressure we are seeing across our three acute hospitals is showing no signs of easing.
"In fact, the pressures on our hospitals are as severe as at any time in the whole pandemic.
"We continue to see exceptionally high numbers of people attend A&E, including many who attend for conditions that would be best treated elsewhere.
"For example, those with sunburn or heatwave-related conditions can access NHS Inform website for advice, or if they need to seek medical attention call NHS 24 on 111 where they will be directed to the right service.
"Covid numbers in our hospitals are rising and this is an additional pressure while we are trying to recover services and treat patients who have planned operations.
"Pressures on our staff is also a concern and they are struggling to cope. The increase in Covid cases is having an impact on staffing levels, with health staff having to self-isolate due to contacts outside of work.
"We have some staff isolating with Covid as well as those on annual leave.
"All our staff continue to work extremely hard to ensure that patients are seen and treated as quickly as possible."
The health board said that due to the pressure on services further planned surgeries have been cancelled.
Ms Park said: "Unfortunately, we have had to cancel a small number of planned surgeries over the last three weeks.
"This is not a decision we take lightly and I would like to apologise for any upset caused."