Coupling cellular communication networks with vehicular ad hoc networks (VANET) can be a very interesting way out for providing Internet access to vehicles in the road. However, due to the several specific characteristics of VANETs, making an efficient multi-hop routing from vehicular sources to the Internet gateways through Long Term Evolution (LTE) technology is still challenging. In this paper, an Internet mobile gateway selection scheme is proposed to elect more potential vehicles to behave as gateways to Internet in VANETs. Therefore, the discovery and the selection of route to those mobiles gateways is carried out via an efficient multiple metrics-based relay selection mechanism. The objective is to select the more reliable route to the mobile gateways, by reducing the communication overhead and performing seamless handover. The proposed protocol is compared with one recent protocol based on packet delivery ratio, average end-to-end delay and overhead. The results show that the proposed protocol ameliorates significantly the network performance in the contrast of the other protocol.