Six local authorities, including Edinburgh and Scottish Borders, have sought £1bn.
Highland Council has applied for up to £300m to support projects in Inverness and the wider Highlands.
Known as City Region Deals, the money would be provided by the UK and Scottish governments.
Similar deals have previously been secured by Glasgow and jointly by Aberdeen and Aberdeenshire councils.
The idea behind the initiatives, which involve direct funding and greater borrowing powers, is to support economic growth.
The Edinburgh and South-East Scotland City Region Deal bid aims to secure major funding to help better protect historic attractions in Scotland's capital, and also boost the wider area's overseas trade and opportunities for new businesses.
Those involved have calculated that an additional £3.2bn worth of private sector investment could be leveraged if the bid is successful.
Highland Council would use its funding for infrastructure projects and jobs creation.
The local authority and the MP for Inverness, Nairn, Badenoch and Strathspey, Drew Hendry, said they expected confirmation of the funding.
The region has already secured £3m of City Region Deal funding.
Scottish Secretary David Mundell confirmed this support during a meeting with Highland Council last year.
The local authority plans to use the money on improving the appearance of Inverness Museum and Art Gallery, upgrading the grounds of Inverness Castle and on creating a free public access wi-fi scheme for the city.
More than £1bn of government funding for Edinburgh, south-east Scotland and the Highlands is expected to be confirmed later in the Budget.