Both concepts seek to use this life as a means to move closer to a supreme Being. For the Hare Krishna movement, the fervent and active devotion to Lord Krishna helps to move individuals in this life towards the realm of Krishna. This takes on the form of intense preaching and devotion to Krishna. The idea is that this life can be used to move one closer to the realm of the divine. Transcendental meditation occupies much of the same domain. In this concept, the individual seeks to surrender themselves in both body and mind to something larger than themselves. The concept of "transcendental" meditation indicates that meditation can be used to "transcend" what is here and what is in the present tense to submit into something larger, a configuration more wide ranging. In this, the concept is driven by the idea of seeking to move beyond what is in front of the individual and direct into a realm that is more embracing of this larger configuration. I think that this is where the two ideas might find convergence. Meditation is part of the Hare Krishna movement, but in a larger sense, both seek to connect the individual to something beyond themselves.
In Hindu religion, there are Vedas and the second part of it is called 'Nigama' consists 'Upanishads' and 'Aranyaka'. There are so many lessons on how to have control over our instincts and habits cuses harm to the body and others. Krishna Dwaipayana Vyasa wrote them all. He thought that it would be difficult to remember them all and so he made out of them some short 'Sutras' which are called 'Vedanta Sutras'. They all refere to the creator of this universe and these lessons teach us to submit to Him fully with our body, mind and speech. Because these are the three thing in a human body which causes lot of trouble to others if not controlled or affected by the vices of others.
In later period, so many saints made illustration of these sutras- Shankaracharya, Ramanucharya, Madhvacharya, Ballavacharya, Nimakacharya and so on. Hare Krishna Movement is spead by Prabhupad Abhayacharanabinda base on the last scripture written by Vyasa that is the Srimadbhagawatam and the Srimadbhagawatagita. Both of the scriptures tell us to submit to Lord Krishna who is the source of the energy of this universe who is the only generator, operator and destroyer of this universe. He is called 'Krishna' because whoever submit to Him blessed with happines. It teaches that one cannot get happines without surrendering to Him as He is the whole and sole of this universe. Real happines is assured in sacrifice of our bad habits rather than in aspiring more and more as he has bestowed everything to us what we need. It also teaches us to become self contented not greedy like the dogs, pigs, camels and donkeys.
In these scriptures it is also taught about the consequenses about afterlife. Hence transcendental meditation gives happines if done truely.