ScotRail said a “small number” of its class 385 electric trains – which run on routes such as the main Edinburgh-Glasgow line – were affected.
It said a “handful of services” would run with half the normal carriages on Monday as a result.
Some of the fleet of 70 Hitachi-built trains, which are less than three years old, are undergoing further checks.
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It follows Scotland-London operator LNER advising pasengers not to travel for a second day after cracks were found in its new fleet of Azuma class 800 trains.
LNER said some Azumas had returned to operation and were providing a reduced service between Edinburgh and London, but disruption was expected to continue “into this week”.
TransPennine Express, which also operates cross-Border trains, has also been affected.
The cracks have been found in the chassis under the carriages, which, in a worst case scenario is thought could cause a derailment.
They relate to a lifting jack used for moving trains during maintenance, which could shear off.
A separate fault involving shock absorbers on trains built by Spanish manufacturer Caf has led to the entire Caledonian Sleeper fleet being checked as a precaution, but operator Serco said no problems had been found.
Disruption to ScotRail services on Sunday was avoided by a strike by conductors and ticket examiners called by the Rail, Maritime and Transport union (RMT), which has forced the cancellation of most trains.
The company said checks on the class 385s, which run on several routes across the Central Belt, should be complete today and most would be able to return to service.
It said the vast majority of the fleet had been found not to have cracks.
A spokesperson said: “Hitachi notified us of the faults with a small number of their Inter City Express trains used by other operators [LNER and Great Western Railway].
"As a precaution, Hitachi has undertaken an inspection of all our class 385 trains.
“There will be no cancellations on Monday, but a handful of services will run with fewer carriages as a small number of the trains need further inspection.”
These will operate with three or four carriages rather than the normal seven or eight.
The spokesperson added: "No 385s were scheduled to be in service on Sunday due to the RMT’s strike action."
A Hitachi Rail spokesperson said: “Inspection has identified cracks on the lifting points under the carriage of some class 800 trains.
"Safety is our number one priority and as a precaution, the decision was taken to halt the entry into service of our intercity fleets pending inspection.
"We are working as quickly and safely as possible to investigate the issue across the remainder of the fleets.”
An LNER spokesperson said: “Due to a number of class 800 trains from several train companies being taken out of service for checks as a precautionary measure, there will be a reduced number of services in operation across LNER.
"The issue is being investigated and once trains have been checked, we hope to be able to release them back into service as soon as possible.
"We apologise for the disruption caused and are advising customers not to travel on Sunday, with tickets valid until Sunday May 16.
"Alternatively, customers can claim a refund from their point of purchase.”