The worker, who did not want to be named, says he often needs to wait to pick up or pack orders in "cramped aisles" at the facility instead of squeezing past colleagues and breaching social distancing rules - but his managers have been pressuring him to work faster in recent weeks.
He says this heightens infection risk and is especially concerned because he has a health condition which puts him at more risk to the virus.
Tesco confirmed that a small number of colleagues tested positive for Covid-19 and are self-isolating and that their close contacts have been informed, as have all colleagues at the distribution centre in Carnegie Road.
But this worker says he only learned about the outbreak after reading an Edinburgh Evening News article last Thursday and feels his supervisor should have called or messaged him before this point. He claims management were then "dismissive" when he alerted them to his health condition in a message querying how safe it was to return to work.
The worker, who has been at the facility for several months through a recruitment agency, said: "I have got vulnerable family members and all they said is 'it's safe to come back, we have measures in place.'
"They have measures in place but it's more that they are not being enforced. When I started there they gave you time to pick up orders but, after a while, they would be getting on to us for not working fast enough."
A Tesco spokesperson said they have been working closely with the union and health and safety representatives and that measures are in place to ensure workers stay two metres apart throughout the centre and that hand-sanitiser, masks, gloves and visors are available to all staff who want them.
All new starters at Tesco distribution centres now receive an induction detailing Covid-19 related safety measures on site.
Social distancing floor markings are used and shift times are staggered to minimise the number of people starting and finishing shifts at the same time.
Equipment in the centre - which supplies about 318 stores across Scotland, Northern Ireland and North-east England – is also cleaned regularly and anti-bacterial wipes are available for colleagues to clean equipment.
The worker acknowledged the measures but says too many people "don't bother" with them and fail to stick to social distancing because they need to keep up a higher pace of work.
The Tesco spokesperson added: "We take the health and safety of our colleagues extremely seriously and are working closely with public health authorities.
"We have measures in place to ensure colleagues stay 2m apart throughout our centres and hand-sanitiser, masks, gloves and visors available to all colleagues.”