It is among schemes across Scotland being considered for the next five years, such as speeding up journeys.
The Borders Railway scheme would involve overhauling a single-line junction where the line to Tweedbank meets the main line at Portobello in eastern Edinburgh.
Supporters of the line have complained about the number of late, overcrowded and cancelled trains since it was opened in 2015. Simon Walton, chair of the Campaign for Borders Rail, said: “That junction is the single most pressing improvement that can be made to the Borders Railway.
“It is a bottleneck and something we have been lobbying for since the line opened.”
Elsewhere, faster journeys between Scotland’s cities, and extending electrification from Dunblane to Perth are being considered.
More space for extra trains in the congested stretch west of Haymarket station in Edinburgh, and upgrades of the crowded Glasgow to East Kilbride and Barrhead lines are among other possible projects.
Improvements to the Inverness to Wick and Thurso lines, and the Glasgow-Oban/Mallaig routes are also on the list.
The proposals come on top of confirmed schemes to be completed by 2024, such as new stations at Reston and East Linton on the east coast main line south east of Edinburgh.
Mr Matheson told the Rail North of the Border conference in Glasgow: “Transport Scotland is working with rail industry partners to develop a suite of projects for consideration, which will cover the length and breadth of Scotland.
“Despite the financial pressures imposed by the UK government, we have confidence in the future of rail.”
The projects would be in addition to a 21 per cent increase in spending on the day-to-day running of the Scottish network to 2024.
The inter-city routes improvements would also be in addition to current work to reduce journey times between Perth and Inverness, and Aberdeen and Inverness.
That work is being accompanied by longer diesel trains to reduce overcrowding.
ScotRail Alliance managing director Alex Hynes said; “We are going to recreate a genuine inter-city network for Scotland.”
Scotland led Britain in passenger growth in 2017-18 with a record 102 million journeys.
The plans follow the electrification of the Edinburgh-Glasgow main line, and routes between the cities and Dunblane. Electric trains are due to run on the Edinburgh-Glasgow line via Shotts from May.