In search of an explanation for the continuing high intensities of ~1 MeV particles observed at the V1 spacecraft over an extended time period from 2002 to 2005 as this spacecraft was moving outward in the heliosphere from ~85 to ~95 AU, we will explore the possibility that V1 did first have a near encounter with the heliospheric termination shock (HTS) at about 2002.6 (assumed to be 1.0 AU inside the shock) and has been moving outward in close correspondence with this shock for the ensuing 2.5 years. We use data from the Massachusetts Institute of Technology plasma experiment on the V2 Web site to calculate the solar wind ram pressure and then use this pressure to calculate the location of the HTS based on pressure balance with the interstellar plasma and fields. We quantify these pressure changes in terms of short term 26-day variations plus longer-term 6--12 month variations. We then use earlier estimates of the response of the HTS to both the long-term and short-term pressure variations. From this we construct a time history of the nominal HTS distance near the solar apex. This distance is found to vary from ~85 to ~100 AU between 1978 and 2005 when we normalize this distance to the location of V1 of 85.5 + 1.0 AU at 2002.6 (e.g., V1 is just 1 AU inside the shock at this time). Specifically, the HTS distance is found to move outward from ~85 to ~95 AU between 2002 and 2005 at an average speed just slightly larger than the V1 spacecraft. Thus if V1 did come close to the shock at 2002.6, it will have moved outward in close proximity to it in the ensuing time period. Short-term variations in this shock distance of ¿2--3 AU, occurring when pressure waves observed at V2 later encounter the HTS, mean that it is possible that V1 could have had several close encounters with this shock during this time period. After ~2005.0, V1 should have, for the first extended period of time, moved beyond the HTS and be moving further away from it.