Two-year-olds Rhys and Shaun Scott died after the incident at the rented property in Dalgety Bay, Fife, on Saturday morning.
Their parents have since told of their shock and devastation at the loss of their “little soldiers”.
An HSE spokeswoman said they were aware of the incident and are liaising with Police Scotland.
She said: “We were contacted by police following the incident and we are liaising with them to try to establish whether there’s any call for the HSE to be involved going forward.”
The boys were treated by police officers and paramedics at the scene before being taken to Victoria Hospital, Kirkcaldy, where they died.
Inquiries to establish the full circumstances are ongoing and officers have been supporting the family.
The scene was cordoned off with police tape over the weekend, with officers standing guard at the front of the home.
It is understood the tank is at the back garden of the house and had reportedly been formerly used for keeping koi carp.
The “long-awaited” boys, whom the parents described as “miracle babies” were reportedly born prematurely and by IVF.
In a statement issued through police, the twins’ parents Mervyn Scott and Sarah Aitken said: “Both boys were long-awaited little soldiers who were full of love, happiness, fun and cheek.
“Individually and together they have touched the lives of everyone they met.
“We along with our family and friends are still in shock and devastated at such a tragic loss from such a ‘freak’ accident. They have been taken from us far too early and will be forever missed and loved.
“We wish to be left alone and be given the privacy needed to grieve.”
A Police Scotland spokesman said: “We are continuing with our inquiries into the tragic deaths of Rhys and Shaun Scott at an address in Dalgety Bay on Saturday 12 March.
“In this instance, we have consulted with the Health and Safety Executive as part of our ongoing investigation.”