The confirmation comes after Health Secretary Matt Hancock revealed earlier this week that the variant had been discovered in more than 80 council areas.
Shropshire Council's public health director, Rachel Robinson, said that two cases had been identified so far.
She said the cases discovered in the area were linked to other clusters in the Midlands, but added that no others had yet been reported.
Mrs Robinson said the positive tests were a reminder of the need to continued to proceed out of lockdown with caution, and for people to get vaccinated when the option is made available.
She said: “A new variant of Covid-19, first identified in India, is being closely monitored in the UK. The variant has been identified as more transmissible, with cases rising across the country.
"Hotspots have been identified predominantly in the North West, but there has also been a small cluster of cases sprinkled across the West Midlands.
“We have had two confirmed cases in the Shropshire Council area linked to clusters of cases out of the county, elsewhere in the Midlands. The individuals and their contacts have followed guidelines and isolated.
"No further cases have been reported at this stage.
“With restrictions easing, this is a reminder that the virus is still with us and we need to proceed with caution. We are continuing to encourage everyone to test twice a week and get vaccinated when you are called.”
It comes as an expert said that the easing of lockdown restrictions should not be affected by the variant.
Dr Arthur Hosie, a microbiologist at Staffordshire University, said the variant may be more transmissible, similar to the Kent variant that emerged in December.
But that he does not think the Government’s roadmap needs to be be altered.
He said: “There is concern that this has increased in dominance in India rapidly and there is preliminary data that suggests there has been a rapid rise in the UK. However, it is hard to unpick if this is due to a change in virus biology or human behaviour.
“The delay in implementing travel restrictions, placing India on the red list, will inevitably have impacted here. But it may be that it is more transmissible, much in the way the Kent variant is.
"Finding this in the Midlands is not a particular concern but does serve as a timely reminder of the need for continued vigilance, social distancing and testing. I do not think the Government will alter the road map.
“I hope, however, that they urge caution and reinforce the key public health messages about the importance of ventilation, wearing of masks and social distancing.
"I would hope that surge testing will rapidly be put in place to suppress transmission where these variants are found.”