It is the latest delay to the Llandeilo route in Carmarthenshire, after the Welsh Government initially said the project would start by the end of 2019.
In a letter seen by BBC Wales, Transport Minister Ken Skates blamed delays on consultation objections by the walking and cycling group Sustrans.
The area’s assembly member Adam Price said he feared the project was at risk.
The Carmarthen East and Dinefwr AM said the letter was “a kick in the teeth to Carmarthenshire residents”.
A £50m deal to build the bypass was agreed between Plaid Cymru and the Labour Welsh Government as part of budget negotiations in 2016.
Mr Price said he would now be seeking an urgent meeting with the government over the issue.
“On no occasion have ministers suggested the bypass wouldn’t be honoured, but that now appears to be a real risk,” said Mr Price, who is the party’s leader.
The main road through the narrow centre of Llandeilo is used as a route from the south to join the A40 through to mid Wales and the Midlands.
Dozens of lorries pass through daily. Air pollution levels are higher than national standards and local people say the problem is getting worse.
They have been campaigning for years for a new route around the town.
In a statement, Welsh Government officials said the current plans for the bypass were being assessed against guidelines.
“The next stage will be the Stage 2 public exhibition and consultation, which we expect will take place by April,” they added.
“Subject to the findings of the consultation, draft recommendations would be made by the summer.”
In a statement, Sustrans said the Llandeilo project was not ambitious enough to improve the safety of pedestrians and cyclists in the town.
The advocacy group added that instead of investing large sums of the Welsh Government budget into new roads, more should be invested in public transport and cycling infrastructure.