The two firms will handle a total of more than $2.5 billion.
The retirement association conducted a search to potentially replace Western Asset Management.
The plan added the core-plus fixed-income asset class earlier this year.
The plan is looking to hire one firm to replace two existing fixed-income managers.
The search follows a structure review that calls for two core and core-plus managers.
The union pension plan is seeking core or intermediate government/credit managers.
The city has issued an RFP seeking investment advisory services for its short- and medium-term investment portfolios.
The retirement fund swapped out its core fixed-income mandate with Western Asset Management at a meeting last month.
The pension is the latest to terminate the fixed-income manager in recent weeks.
The plan’s current manager has been under scrutiny due to organizational and personnel changes.