Fellows was chosen by the hiring subcommittee during a meeting Wednesday from a field of eight candidates, according to an e-mail that Senate President John Valentine sent to lawmakers.
The recommendation will be voted on by legislative leaders next week.
"I have grown to trust and have great faith in his legal abilities, management skills, problem solving ability, people skills and vision for the legislative institution," Valentine wrote.
"The Speaker [of the House] and I have complete confidence that Mr. Fellows will carry on the tradition of strong, non-partisan legal counsel to the state legislature."
Fellows has worked in the general counsel's office for 22 years, most recently as deputy general counsel. He earned his law degree from the University of Utah College of Law.
"Working with John for the past 14 years has been an absolute privilege. He is a skilled and passionate attorney but more importantly he is a zealous advocate for the legislature and it's processes," House Speaker Greg Curtis said in a statement.
Taylor is leaving the office at the end of the month. She was hired as associate general counsel in 1983 and promoted to general counsel two years later.
The general counsel is the nonpartisan chief legal counsel for the Legislature and its committees, drafting bills, representing the body in court cases, and preparing legal opinions on legislation - although the Legislature no longer makes the opinions public, on Taylor's advice.
Taylor also is the staff attorney for several top legislative committees and a member of the National Conference of Commissioners on Uniform State Laws and the Legal Services Staff Section for the National Conference of State Legislatures.
During her tenure, the staff at the office grew to 14 attorneys and handled major changes to tax, elections, human services, retirement and other sections of the law.
She had a hand in writing several controversial measures at the Legislature's request, including the state's strict abortion law, school gay club prohibitions, union paycheck protection legislation, and school voucher bills.
