Hearts are favourites to sign the Northern Ireland internationalists Kyle Lafferty and Michael Smith.
They will hold a third day of talks with Lafferty on Tuesday, but Edinburgh rivals Hibs also want to speak to the former Norwich City forward.
Hearts discussed terms with Lafferty during lengthy talks at Tynecastle on Monday, which followed an initial meeting on Friday. They made it clear they won't break their wage structure but are prepared to make the 29-year-old one of their highest-paid players.
He is due back in Edinburgh for further negotiations on Tuesday as he seeks a move back to Scotland, where he spent four years with Rangers between 2008 and 2012.
Hearts are keen to win the battle for his signature and parade him before the week is out. Hibs, though, plan to usurp their city neighbours if they can meet the player face to face this week.
They want to tempt Lafferty to Easter Road after losing Jason Cummings, James Keatings and Grant Holt from their forward line since the end of the season.
Right-back Smith was shown round the facilities at Riccarton on Monday. Hearts have agreed a transfer fee with his club, Peterborough United, and the 28-year-old believes a move to Scotland could help him add to the Northern Ireland cap he won last year.
He captained Peterborough at times last season but was transfer listed at the end of the campaign despite playing 44 times for the English League One club.
Coventry City had an offer accepted for the attack-minded full-back, although he chose to travelled to Scotland after hearing of Hearts' interest. Both he and Lafferty are interested in working with the Edinburgh club's assistant coach Austin MacPhee, who is also part of the Northern Ireland coaching staff.
MacPhee is understood to have been instrumental in getting the players to Edinburgh, where they will join their international colleague Aaron Hughes if they put pen to paper with Hearts.