The Spanish club have a strict policy of signing players native to the Basque Country or neighbouring regions - and the aforementioned Premier League players are two who fit the bill.
Arrutia has also declared Athletic Bilbao's interest in Real Madrid midfielder Asier Illarramendi and Atletico Madrid's Raul Garcia.
Athletic Bilbao confirm interest in players
Athletic's club president was quoted by Spanish newspaper AS - according to the Daily Mirror - as stating: "There was interest from Athletic in players, although it was focused on Illarramendi. There are not so many players: [Asier] Illarramendi, Raúl García, [Nacho] Monreal, [César] Azpilicueta.
"We have to be aware, monitoring– whether they continue at their clubs [or not], it’s because they are interesting."
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Monreal unlikely to leave Arsenal
Monreal has been in and out of Arsenal's starting line-up since completing his £8.3 million move to the Emirates Stadium from Malaga in 2013.
The Spain international is competing with Kieran Gibbs for the left-back position, although he has played at centre-back on occasions this term.
Monreal has started in all three of Arsenal's most-recent Premier League fixtures - including the shock 2-0 victory over Manchester City at the Etihad Stadium - and is unlikely to be sold by Arsene Wenger this summer.
Azpilicueta a key Chelsea star
Athletic Bilbao, however, stand even less chance of prizing Azpilicueta away from Chelsea.
Jose Mourinho's admiration for Azpilicueta is no secret - he once claimed Chelsea would win the Champions League with 11 Azpilicuetas on the pitch - and the fact the Spain international can play at both left and right-back makes him an extremely useful asset for the Blues.
Widely regarded as one of the best full-backs in the Premier League, it would take an astronomical bid to even tempt Chelsea into negotiations over the 25-year-old's sale.
Athletic Bilbao might have more luck with their pursuit of Real Madrid's out-of-favour midfielder Illarramendi - who is also wanted by Arsenal and former club Real Sociedad - but Garcia is extremely unlikely to swap La Liga champions Atletico for San Mames.