United Lincolnshire Hospitals Trust has lobbied the Department of Health and Social Care for capital funding in order to carry out the work.
Officials said the trust needs between £15 and £20 million for the plan, similar to that secured for Boston Pilgrim Hospital last year.
It follows the significant pressure at the county’s emergency departments in recent months which has seen demand soar at the county’s hospitals.
Mark Brassington, chief operating officer at ULHT, said the trust’s A&E units are “not the size they should be” in order to deal with the volume of patients.
“The key thing that we need to address is that the departments that we currently have on all of our sites are not the right size for the level of demand that is coming through,” he said.
“We know that we need to increase the size, but secondly we need to improve the environment so that we can ensure that every time we treat a patient it’s with privacy and dignity respected at all times.
“At the minute, we are providing care to patients in environments that we would not necessarily wish to because of the sheer volume of patients that we have in our departments.”
Care Quality Commission inspectors revisited both Lincoln and Boston A&Es earlier this month to carry out winter assurance inspections.
Concern was raised by the CQC that departments were still under pressure, but inspectors acknowledged that improvements were being made.
Mr Brassington said the trust has seen pressure at A&E from both walk-in patients and those who arrive by ambulance.
It has also seen more people being sent to Lincoln since the decision was taken to close Grantham Hospital’s A&E at night.
He added that the organisation’s performance for its emergency departments is “not where we want to be”, but added that it was “improving” and that the units were “safe”.
The trust has failed to meet the four-hour waiting time target for A&E for more than half a decade. ULHT last met the target in September 2014.
The national standard for A&E waiting times in England is 95 per cent. The average from across the country was 79.8 per cent last month.
NHS England data from December 2019 showed the number of patients waiting less than four hours in Lincolnshire was at just 64.7 per cent - well below expected levels.
In the past few months, patients have been told to avoid Lincoln A&E due to it being “exceptionally busy” and GPs increased the number of emergency appointments to help cope with demand.
ULHT saw a “significant increase in very poorly patients” at the department in November, with some waiting as long as 12 hours to be seen.
In an effort to ease the pressure, hospital bosses want to extend the units at both Lincoln and Boston.
Following a visit from Prime Minister Boris Johnson last August, a total of £21.3 million worth of capital funding was secured for Boston A&E.
It was part of an £850 million fund which the Prime Minister announced for 20 hospitals in the country.
Trust officials said the department at Boston is half the size it needs to be in order to meet its demand.
The money will go towards improving patient flow at the unit and setting up “care zones” to direct people who arrive at A&E in an effort to improve waiting times.
This would allow patients who need emergency treatment to be seen and treated quicker.
Mr Brassington said a similar proposal has been put forward for Lincoln A&E, however there is no indication of when the government will make a decision on the plan.
Carl Macey, chairman of the Health Scrutiny Panel for Lincolnshire, said the committee supported the decision to expand the department.
“We understand they are under pressure, we understand there is difficulty in recruiting staff whether it doctors, nurses and so on,” he said.
“However, we want to see that there is some positive progression within the urgent and emergency care departments.
“Any improvements that could be made for the residents of Lincolnshire we would fully support.”
Officials at ULHT have called for patients to only attend A&E in case of emergency.
A spokesperson for the trust said: “A&E should only be used for life threatening emergencies and we would ask that people choose well and consider going to their pharmacist or GP, calling NHS111 for advice where appropriate or using the ASAPLincs website or app.”