The petition requests a special meeting of University Council to discuss whether faculty still has confidence in the senior administration.
The faculty claim non-academic positions have increased while teaching jobs have been cut, that education is being hurt and that donor funds are being mismanaged.
A special meeting requires a petition signed by at least 50 members of the council and there are 38 signatures so far.
The faculty says in a letter to media that it would be inappropriate to talk about the petition at such an early stage.
Lee Elliot, chairman of the university's board of governors, declined to be interviewed, but said in a written statement that the board has confidence in the school's administration.
"The board of governors appreciates council members' continuing interest in the well-being of the University of Regina. At the same time, we would like to express our confidence in the way the president and members of the executive team are managing our university and helping fulfil our collective academic mission," read the statement from Elliot.
A document accompanying the petition outlines the concerns of the faculty members.
It says: "Reduced academic resources in core faculties of the university have diminished the research profile of the university, reduced the number and range of courses we offer, reduced the quality of our programs, and thereby reduced the quality of the education we provide."
It claims the president and academic vice-president "are responsible for a significant portion of these damages."
The faculty members also say there are questions of transparency over how money is spent. They allege $1 million for the creation of a trust fund to support a research chair with a focus on heavy oil recovery processes was instead used to cover overspending in the faculty of engineering.
If there is support for a special meeting and University Council passes the motion, a non-confidence vote could be held by secret ballot.
The University of Regina has been struggling to balance its budget and facing controversy involving a carbon capture centre.
The school had hoped to see an operating increase of between four and five per cent in the budget tabled by the Saskatchewan government in March. It got about two per cent.
University president Vianne Timmons said at the time that departments had already been asked to cut three per cent from their budgets and there would be "some tough slugging ahead."
Students are also being hit in the pocketbooks. Tuition fees for undergraduates are going up 4.4 per cent, while masters and PhD students will pay 10 per cent more, except for those in the graduate schools of business and public policy.
In May, Saskatchewan's auditor told the University of Regina to fix its policies for research projects.
Bonnie Lysyk made 26 recommendations on oversight, updating policies and procedures, evaluating risks and monitoring compliance.
The university asked the auditor to do the review in large part because questions were raised about how more than $2 million was spent at the International Performance Assessment Centre for the Geologic Storage of Carbon Dioxide, commonly known as IPAC-CO2.
An internal IPAC memo said the greatest part of the money billed for IT services was "spent for no acceptable business reason."
The centre was based on the university campus until the province cut funding earlier this year and it wound up operations.
Timmons said May 30 that the audit was also about boosting public confidence in research funding after the IPAC controversy. She said the school accepts the auditor's suggestions and hopes to implement them within 18 months.