Facebook splashed more than $1bn to acquire Instagram in April, after the mobile service gained millions of users on iPhone and Android devices.
Currently, there is only basic functionality on the Instagram website, allowing users to download the mobile apps or change certain elements of their profile, such as their password.
Any Instagram images can be viewed and shared via weblinks, but users are unable to view their full profile pages on desktops or laptops.
But TechCrunch reports that web designer Cole Reinke claims he was recently logged in to the web version of Instagram and saw a new tab called "view profile".
The tab led to a '404' page with his Instagram username, and then disappeared entirely, but Reinke believes that this suggests work is being done to take the service onto the web.
Instagram allows people to capture photos and then apply a range of filters to the images, before sharing them with their contacts on the service.
Kevin Systrom, the co-founder of the company, previously said that mobile is Instagram's primary focus, and refused to discuss a potential web offering.
Asked by Time magazine about the issue, he said: "We are fully sold on the fact that if we're going to do big, meaningful stuff we need to take on networks. We believe mobile is that."
He added: "You can't take a desktop experience and shove it into a 3-by-4-in screen. It's a very different behaviour pattern.
"It's a very different browse pattern. People interact with their phones very differently than they do with their PCs and I think that when you design from the ground up with mobile in mind, you create a very different product than going the other way."
However, Instagram has always said that its web presence is "still developing", and so an extension of the service could be on the cards.
Certainly Facebook will want to maximise exposure of the service to justify its huge outlay on the acquisition, which is still in the process of going through regulatory approval.