Osborne was due to start work on the new Shoreham Footbridge this spring.
But the project is on hold because “current estimates from the contractor suggest the cost of the new bridge will significantly exceed the project budget.”
West Sussex County Council is now working with Osborne to review designs and costs in a bid to “achieve the earliest possible start date.”
County Council Cabinet Member for Highways and Transport, Pieter Montyn said: “The new footbridge will cost a significant sum of money to construct.
“The County Council has a responsibility to ensure that we get value for money. Therefore, it is important that sufficient time is taken to check and agree the contract price for the construction of the bridge.
“This is tax-payers money at the end of the day and it must be spent wisely.
Funding for the project has come from the County Council and sustainable transport charity Sustrans, which received a £50 million grant from the Big Lottery Fund to extend the National Cycle Network into towns and cities across the UK.
Sustrans Regional Director for the South East, Simon Pratt, said: “Obviously it’s disappointing that project costs have increased.
“We’re keen to continue to work with all partners to find a way to make this project a reality and provide local people with more choice when making everyday local journeys.”