The offer to pay 55p a share is being recommended by the WYG board to shareholders. It offers a premium of more than three-times the 16p share price.
A deal would give the US firm, which employs around 18,000 staff out of 400 locations worldwide, access to the UK and European markets.
WYG employs around 1,600 people out of 30 locations across the UK, Europe, Africa and Asia, but the £150m revenue group has been struggling with losses.
Earlier this year WYG shares tumbled by 40% after the board issued a profit warning saying the UK market was being hit by project delays and deferrals across the public and private sectors.
WYG also said it was in danger of breaching its banking covenants and had opened discussions with its lending banks.
Jeremy Beeton, non-executive chairman of WYG, said: “The WYG board and I unanimously recommend the offer from Tetra Tech.
“Becoming part of the Tetra Tech Group enables benefits of scale and access to expertise across highly complementary geographies and client relationships, and brings operational infrastructure and financial strength to support WYG’s long term growth ambitions.
“We provide Tetra Tech with a strong platform for investing in growth in the UK and Europe. We believe the combination of our two companies will provide greater opportunities for our people and clients, as well as offering certainty of immediate value to our shareholders at a considerable premium to the prevailing share price.”
Dan Batrack, Tetra Tech chairman and CEO, said: “WYG advances our strategy to add firms that support our position as the premier global high-end consulting, engineering, and program management firm.
“WYG’s expertise in infrastructure and program management, as well as water and environmental services, enables us to deliver innovative solutions to support the UK’s infrastructure needs.
“Together, we will be able to provide an expanded scope of services to our customers and offer our combined staff even greater professional opportunities.”
Tetra Tech said it expected to make cost savings across the business through modest cuts in back-office overhead support and corporate management activities across the combined group, to boost the profitability of WYG.