The operator said 88 posts were “at risk” and routes in the area had not been viable for several years.
It comes four years after First significantly reduced its services in East Lothian with the closure of its Dalkeith depot.
Lothian Buses replaced services on several routes.
In all, First will axe 35 services, including routes to Galashiels in the Borders and Stirling.
First Scotland East managing director Paul McGowan said: “I’m disappointed that we’re proposing to withdraw from Musselburgh and North Berwick.
“We will today engage with the local authority on the future of bus services.
“However, given there are other bus operators that operate in and around the area, I’m confident if we decide to go ahead with these proposals the majority of our customers won’t be left without a service.”
Lothian Green MSP Alison Johnstone said: “The potential loss of these jobs is terrible news for the individuals and families concerned and for the wider East Lothian economy.
“Public transport needs radically improved in East Lothian with many local people dependent on buses and often overcrowded trains for work, healthcare and social journeys.
“Today’s announcement shows the danger of leaving such a vital public service in the hands of private firms.
“The withdrawal of First bus services could leave a serious gap, and I would urge East Lothian Council and Lothian Buses to look at ways to provide those services.”