Tim Cahill, whose 11 Premiership goals last season were an important factor in Everton's unexpected qualification for the Champions' League, has not had a full summer break this year or last because of playing for Australia and Moyes believes the fatigue is beginning to show.
As Portsmouth achieved their first away win since Christmas at Goodison Park, Everton were collectively unimpressive. Moyes was anxious not to attribute blame to one individual over others but he agreed that Cahill's latest trip to the other side of the world appeared to have taken its toll.
"It would be wrong to single out Tim Cahill after today's performance because there were a few who had a tough afternoon but I am beginning to look at him with a bit of concern," Moyes said. "It is difficult, to be fair, to have flown back from two games in Australia and then have to play again within a couple of days but he is paid by us and you expect his performances to be for us.
"You want to have him in the team because he can score goals," Moyes added, "and if he had taken a couple of the chances he had against Manchester United and Villarreal then maybe we would be looking at him differently."
There is little he can do to stop Cahill clocking up another air-miles marathon in November when Australia play their World Cup play-off. Of more immediate concern is Everton's record so far of three points and only one goal from four games.
Portsmouth were a side, Moyes admitted, that they expected to beat but the south coast visitors showed the greater imagination after giving debuts to Salif Diao and Dario Silva and with a little more composure close to goal might have won more handsomely.
Everton, for whom Nuno Valente made a laboured debut at left-back, resorted early in the match to lumping long balls towards Duncan Ferguson's head. It was ironic that the Scotsman should ultimately provide the decisive touch at the other end, glancing the ball into his own net in an attempt to prevent Dejan Stefanovic doing so from a 60th-minute corner.
Goal: Ferguson (60 og) 0-1.
Everton (4-4-2): Martyn; Hibbert, Yobo, Weir, Valente; Davies (McFadden, 66), Cahill, Arteta, Kilbane (Osman, 66); Bent (Beattie 66), Ferguson. Substitutes not used: Wright (gk), Ferrari.
Portsmouth (4-4-2): Ashdown; Priske, O'Brien, Stefanovic, Vignal; O'Neil, Diao, Hughes, Robert (Vukic, 72); LuaLua (Karadas, 79), Silva (Viafara, 70). Substitutes not used: Westerveld (gk), Griffin.
Referee: M Atkinson (W Yorkshire).
Booked: Everton Davies, Valente; Portsmouth Hughes.
Man of the match: Silva.