I am an ENT specialist and can help you with this.
You are having enlarged lingual tonsils
Lingual tonsils are lymphoid tissues which are found normally at the base of the tongue, just like the palatine tonsils (which are at the side of the tongue and which we normally mean when we say 'tonsils').
Just like the palatine tonsils, the lingual tonsils can also get enlarged due to infection, inflammation, etc.
You are likely to be having a mild infection which has caused these tonsils to be become enlarged.
Since it is bilateral this is the most likely cause.
Usually gurgle is sufficient - use a mild antiseptic gargle like Betadine gargle, H2O2 gargle or any other antiseptic gargle - gargle regularly about three times a day specially after meals.
The other possibility is that these are circumvallate papillae - these are taste buds at the base of the tongue - these also may become enlarged in the same way - treatment is the same in this case also with gargle.
However, if this does not resolve the problem - see your doctor. A clinical examination will confirm the cause - if necessary your doctor can prescribe prescription strength medicines for this.
Look up here for images for lingual tonsils and circumvallate papillae:Lingual tonsilsCircumvallate papillae
Take care - gargle is sufficient for either usually - but see a doctor if it persists.
I hope my answer has helped you. Please click on the green 'accept' button below for accepting this answer if you have been helped (or click on a 'smiley' feedback if there is no accept button). Feedback/Bonus will be appreciated. Do not hesitate to ask further questions if you need more info.
Dr. P.Jyoti, MD.