The reported Leeds United target's stock has risen recently.
The 31-year-old's stock has risen significantly this month after a fine campaign in France as the Northern Irish bowed out at the last-16 stage by defeat to Wales.
He will be a free agent come the end of June when his contract with Hamilton expires and he is expected to be in demand, with his display in a 1-0 defeat by Germany particularly impressive last week.
HITC Sport reported last week that Leeds United were among those looking at McGovern as manager Garry Monk looks to bolster his goalkeeping options.
Nevertheless, the man himself says that there may be developments this week, and he hopes to have some "good options" on the table.
"Regarding my future, I don't know what's happening because I've been focusing on the Euros," he said via The Belfast Telegraph.
"The games come so thick and fast it's hard to focus on them, never mind the future. This week I'll sit down and hopefully I will have some good options.
"There's been a bit of interest since the Germany game, as there was before the tournament. I wanted to come to France without signing a contract because I thought it was a great opportunity. Obviously Hamilton are in touch as well."
Marco Silvestri was Steve Evans's number-one keeper at Elland Road, and has been since his arrival in 2014, but it remains to be seen how highly new manager Monk rates the Italian.
Nineteen-year-old Bailey Peacock-Farrell might also try to impress in pre-season after being given a taste of first-team action toward the end of last season.