Construction information specialist Glenigan has BAM, Miller, Willmott Dixon, Shepherd, Morgan Sindall and Balfour Beatty in the hunt.
Sources at architect ADP said a winner would be chosen “within the month” with work set to start on site by the end of August.
Completion of the library is due by the start of 2015.
The new library will provide 1,000 reader seats, flexible training rooms, a skills zone, community classrooms and café, plus flexible space for 6,800 linear metres of book stock.
Josh Smith, Leeds University Union’s Education Officer said: “I am delighted to have been a part of the project team for the new library and I am excited for students to have access to what will be a very up to date building to study in and use as a resource.
“Good quality library space is always going to be a priority for students, which is why this new facility is going to be a fantastic addition to campus.”
Dennis Hopper, the Director of Facilities Management at the University of Leeds said: “The end result will be stunning and a visible sign of our commitment to invest in the student experience.”