They are both earmarked for land south of Squinter Pip Way in Bowbrook.
The mainstream primary school, funded by capital funding from Shropshire Council and a special school, funded by the Department of Education as part of its free school capital programme, are now scheduled to open in 2023.
Both schools were planned to open in September 2022.
Construction is planned within the same timeframe, using a single lead contractor.
A re-procurement exercise is currently under way and it is hoped a new contractor will be in place towards the end of the calendar year.
As a result of the delay the opening date for the two schools has been moved to September 2023.
The council, Department for Education and the separate academy trusts who will be operating each of the schools, say they will continue to work in close partnership to ensure that there are no negative impacts on the pupils who will attend the schools.
Councillor Kirstie Hurst-Knight, Shropshire Council's cabinet member for children’s services, said: “We are confident that the delay in the opening of the new mainstream primary school in particular should not cause adverse pressure on the reception intake in September 2022 as there are forecast be sufficient reception places at schools in the west Shrewsbury area next September.
"This is due in part to a forecast dip in the national birth rate, which is reflected in the 2022 known children numbers for schools in the Bowbrook area.
"The new primary school, when opened, will be operated by The 3-18 Education Trust.
“While there is a delay in the building of the special school, we are still focusing our efforts on expanding and enhancing the range of specialist provision already on offer for children.
"Our aim is to ensure that children with special educational needs and disabilities have, and expect, the same opportunities in life as other children and young people."
She said temporary accommodation for a small cohort of pupils who would have attended the new special school in the 2022/23 academic year is being investigated by the Department for Education and the academy trust which will be operating the school – The Youth Engagement Schools Trust.
A decision on the preferred location will be determined in the coming weeks.
“The two new schools are an important and exciting development as we look to meet the demand for school places in this area of Shrewsbury," added Councillor Hurst-Knight.
The special school will cater for children aged five to 16 who have special educational needs in the area of communication and interaction.
It will also cater for those who may have additional needs in the area of social, emotional and mental health, requiring a level of specialist intervention not ordinarily available within a mainstream school.