Bellamy and Arteta are both being linked with switches to and from Eastlands and Goodison Park in Sunday's tabloids.
The former looks to be on his way out of City after failing to strike up much of a relationship with manager Roberto Mancini.
Bellamy has been with six different clubs in the last five years but is understood to be keen to stay in the north-west - a factor that could prove crucial for Moyes who would probably have to shell out around £10million to bring the 30-year-old back to Merseyside, where he spent a season with Liverpool in 2006/7.
Meanwhile, City are said to have targeted Arteta as they prepare to splash the cash once again this summer.
The 28-year-old reminded admirers of his class in the second half of the Toffees' campaign after recovering from a serious knee ligament injury.
The relationship between City and Everton remains frosty following the protracted Joleon Lescott deal last summer, but a bid of £20million or higher could be too tempting to turn down.