Scovill Inc.'s board of directors has unanimously rejected a $35-a-share bid from First City Properties Inc., and has authorized management to explore other alternatives, including the sale of the company.

Scovill, the Waterbury, Conn.-based company that makes Hamilton Beach appliances and Yale locks, received the unsolicited $430 million takeover bid Dec. 20 from the Belzberg family of Canada through First City of Beverly Hills, Calif., which it controls. The offer for all outstanding common shares sent Scovill stock soaring more than 9 points last week, and the stock closed Monday on the New York Stock Exchange at 39 1/2 , up 1.

In a prepared statement, Scovill said its board had determined that the Belzbergs' bid "is inadequate and not in the best interest of Scovill, its stockholders and its employes," and urged shareholders to turn down the tender offer. In rejecting the bid, the board considered the opinion of its financial advisers, Morgan Stanley & Co. and Morgan Lewis Githens & Ahn Inc.

Scovill is vulnerable to a takeover because management owns only a small percentage of the stock and because the company has not adopted antitakeover measures. It is an attractive takeover candidate because of its strong brand names, especially Hamilton Beach and Yale. The company also makes industrial products through its Schrader Automotive Group and Schrader Bellows Automation Group. Its Nu Tone Housing Group makes exhaust fans, intercoms, heaters, door chimes and bath cabinets.

"A strong balance sheet is one virtue this company has," said American Securities Corp. analyst David S. Leibowitz. As of Sept. 30, Scovill had assets of $535.3 million and long-term debt of only $65.4 million.

Scovill had sales of $742.6 million in 1983 and profits of $27.4 million ($2.53 a share).

In the first nine months of 1984, Scovill had sales of $616.7 million and net income of $24.6 million ($2.02) compared with sales of $531.5 million and profits of $16.2 million ($1.55) for the same period of 1983.